I'm just starting a campaign for the first time with a couple of friends, and I've been told that certain classes like cleric are really strong; some people even say they're overpowered. What is it about them that makes them so strong?
You might want to give a look at this article and its explanation, especially the part relative to tier 1 classes. Should the links ever go down, it's the famous tier system for 3.5 classes, with the explanation of why each class is in its tier and a detailed explanation of tier 1, where the cleric belongs.
To make a long story short, a cleric is a full caster, a class that can get access to level 9 spells. While lower level spells ofted deal with inflicting conditions, removing them or otherwise messing with the numerical aspects of the game, the higher you go the more spells start doing things like asking information to powerful entities, downright killing or returning to life people, mind controlling, dishing out immunities and so on. Up to the point of having any effect, as long as your deity agrees.
At lower levels, the cleric is a caster class that has no problem being a fighter type as well. It has spells that let him overcome his inherent weakness (his medium BAB progression), making him one of the best melee characters in the game.
Was this not enough, the first level of cleric is frontloaded with the whole set of class features (access to all the daily uses of Turn Undead and to both domains' related powers, often consisting in bonus feats), making it the best 1-level-dip class according to many.
There are quite a few reasons that Clerics are great. Let's look at some of their class abilities and what they do for you.
Clerics are full spellcasters, with access to a very large spell list. While not quite as expansive as the Wizard list, spells for every situation going up to extremely powerful stuff are on here. Full spellcasting is the single strongest ability in D&D 3.5, and Clerics have it.
Domains tie in with a Cleric's deity, and give you some extra powers. Some of these are pretty minor. Some of them are pretty awesome. They include things like a die reroll once a day, the ability to use Turn Undead as Destroy Undead once a day (which can flat out end Undead encounters instantly if used correctly), Freedom of Movement for free, and so on. If you use more than core books, there are some awesome ones.
Domains also give you an extra spell and spell list, including some options that you can't normally get as a Cleric. Chosen correctly, they can add a lot of versatility.
Clerics are proficient with all armor, and with shields. That's a spellcaster in full plate. They're also proficient with Simple Weapons, but there are prestige classes, domains, and races that let you get some or all Martial Weapons as well. That means they have the tools to do physical combat effectively if you want to.
Clerics have two good saves — Fortitude and Willpower. Those are the two that matter most. They get d8 hit dice, which is pretty solid. They only get 2 skill points a level (which is bad), but they don't need very many skills to be successful. No problem here.
For attributes, Clerics really need Wisdom, Strength (if focusing on melee combat), Constitution, and Charisma (if trying to focus on Turn Undead). Being able to dump several stats is really helpful in optimizing yourself, and they can get down to needing 2–3 stats to be decent.
Absurdly effective against Undead, this can end fights instantly. Can also be made to work against some other monster types with spells or domains.
But it gets really awesome if you get the feat Divine Metamagic. That lets you use Turn Undead to spontaneously apply metamagic feats without boosting spell level. Apply it with a really expensive feat like Persistent Spell, and you suddenly take some of your awesome buffs and make them last for an entire day.
This can get silly very quickly.
Clerics have access to a long list of good prestige classes, several of which give you bonuses while giving up absolutely nothing in your progression. They're just gravy on the top.
Notice how much awesome is on that list? Now what happens if you put it all together?
Clerics Can Beat Any Situation
There's a popular post with a tier-system for classes, that focuses on how good they are and how many different types of encounters they can handle. Clerics are tier 1 (the highest), because there is pretty much nothing you can throw at them that they don't have a way to handle.
Compare to something like a a Rogue, which is pretty good at a few things, and really weak when facing other things. A Cleric is good, no matter what the situation.
When optimized well, Clerics can even be better at other classes specializations. A person in my campaign right now usually plays Fighters, but wanted something with a bit more oomph. It was simple for me to make him a Cleric that's better at being a Fighter than an actual Fighter... and is also still a full spellcaster. That was done with Persist Spell on Righteous Might and Divine Power, giving this Cleric a Fighter's attack bonus, 10 bonus strength, some bonus constitution, and large size (giving reach). That's not a particularly difficult character to make, and nowhere near the most optimized a Cleric can get.
Oh yeah, and he saves money on gear because he can use spells like Greater Magic Weapon to enchant his own stuff. At level 12 he can turn simple masterwork gear into a +3 sword, +3 armor, and +3 shield. That leaves a lot of gold that can be spent on gear for other abilities. Should he get into trouble in melee, he has a full spellcaster's spell list to fall back on.
Clerics are a class with good proficiencies, good stats, good class features, full spellcasting, and the ability to be super versatile. They're good at pretty much everything, and you can never put them in a situation where they're forced to say "oh, I'm totally helpless and need another class to save me." That is what makes them so good.
Setting aside the tier system as it is well articulated in other answers, I'll draw from my personal experiences. To summarize:
Some classes reward system mastery more than others. For the expert player, certain patterns in the rules make anything possible. Spellcasting is one of these patterns.
When I've played, both when I've started playing and up to the time I stopped bothering with 3.5, I observed some people having more success at the game due to both more creativity and more options.
Most any class will reward player creativity, to a point carefully delineated (implicitly or explicitly) by the DM. However, it is the especial purview of pure spellcasters who have their spell lists well internalised and understood who can solve an astonishingly large number of problems with "I've got a spell for that."
Now, to be clear, most clerics I've seen played were forced onto the players playing them. As such, they were heal-batteries with little care for the, quite literally, hundreds of possible spell combinations that they could have. Clerics and druids, because they have all spells available can solve most any problem given sufficient in-game prep time and out-of-game research time. Unfortunately, I've played with few clerics willing to do the out of game prep for more than a single level of spells (especially with the need to establish multiple contingent spell lists for situations your DM specialises in). Furthermore, most 3.5 games I've played in have significant plot pressure combined with (what amounts to) in media res openings. Few of the players I've gamed with were fast enough on their feet to swap out spell lists in reaction to their environment in this sort of game.
The literature available online has a significant system mastery bias: in order to contribute to the literature and have that contribution noted the poster must demonstrate capabilities. As this is a well known domain with no innovation, the level of capability required for discourse is quite high.
Your milage, with a new group, will almost certainly not correspond with the theoretical and practical optimisation discussions online. While yes, given a functionally memorised SRD and Spell Compendium, the Cleric (and druid) are astonishingly capable characters. Until that system mastery is intentionally inculcated, players will have a hard time judging and achieving capability and power level.
In my experience, most clerics I've played with were so intimidated by the huge number of spells available that they found their one spell list and stuck with it. While some of the spells on that list could be good (either via other player suggestion or sheer luck), the versatility that marks the expert cleric is almost always absent in this mindset. They know what they can do, and because they know and feel comfortable with their specifically chosen spell list and little else, they certainly won't display the overwhelming strength common to clerics in high system mastery discussions.
While intelligence agencies (and seemingly theoretical optimisers) are required to assess other state actors by their "capabilities, not intentions;" players in the same game are not required to abide by that standard. Therefore, don't worry overmuch about theoretical claims on the net. While they are certainly true, they are only true after a given level of system mastery and desire on the part of the player. For my own case, they were only true after I functionally served an apprenticeship in optimisation.
At the end of the day, optimisation and relative strengths are a function of group intentions, standards, and effective communication. There are plenty of places to do the necessary readings and interact with the necessary people to serve that "optimisation apprenticeship." For a certain type of person, optimisation is a delightful exercise in building to requirements, expressing a wish for personal power, or otherwise exercising validatory psychological experiences.
It’s the Spells
There are a lot of other things that are really, really nice for clerics: decent proficiencies and chassis, the existence of Divine Metamagic, Persistent Spell, and other divine and metamagic feats, the various Granted Abilities that you can get from the Domains that you choose. These are all nice. A couple of them (Divine Metamagic chief among them) are fairly overpowered, too. But ultimately, these are not what makes the cleric overpowered. Losing them would not make the cleric not overpowered. Less powerful, true. Without Divine Metamagic, in particular, it’s fairly hard to make the cleric “a better fighter than the fighter,” though by no means impossible, and the wizard starts to look a bit “strictly superior.”
But even in a core-only game, with the armor proficiencies stripped, HD downgraded, Fort save reduced, Granted Abilities removed from Domains, clerics are still overpowered.
And that’s because of the spells. Not just high-level spells, either, but spells at every level. Even just in core (actually, especially the core spells), the spells just do too much, are too flexible. It’s an almost impossible problem to fix, because you literally need to go through every spell and figure out A. if this one is one of the problematic ones, and B. if so, how to reduce it. That is an enormous amount of work. I have never seen a comprehensive homebrew attempt to do it. Paizo certainly didn’t do it, or even try to, in their Pathfinder spin-off. To date, the only 3.5 spin-off I’m aware of to even attempt to do so is Legend, and that was because they simply started from scratch with an almost-entirely overhauled system.
But so long as clerics get the spellcasting they do, with their spells, plus Domain lists to expand their repertoire and pick up some more juicy spells that they missed, they are an overpowered class.
This answer will be different from most. While I am well aware I'm in the minority on this, the "tier system" and the supremacy of spellcasting in 3.5 are not beliefs I agree with. In my opinion, the belief in these concepts are the result of "sub-optimal" game mastering, not sub-optimal playing or character design, in that most DMs never learn to build encounters and employ tactics that would reign in spellcaster "superiority," allowing casters to run roughshod over non-casting players. Worse, the common acceptance of the tier system has led many gamemasters to believe this is an unavoidable feature of the 3.5 system, and so not even try to correct it. Its resulted in a vicious circle of reinforcement that has carried over into Pathfinder and other 3.5 based games.
That said... even by my reckoning, the class features of the cleric are the strongest of any class in the game. This isn't because clerics can do everything better than everybody else. If the DM knows what he's doing, a cleric cannot outfight a fighter and he definitely can't outsneak a rogue, at least not reliably or for long.
Clerics do, however, have two edges that push them over the top: armored spellcasting and domain powers.
Despite Mike Mearls' latecomer downplay of it, armored spellcasting is a huge deal in 3.5. For the most part, to a noncaster, defense is just defense. It slows down the rate at which you take damage and lets you fight longer before dying. But to a spellcaster, defense is both defense and offense: it both delays your death AND allows you to finish casting your spells uninterrupted. Remember also that the ability to wear armor does more than just raise AC; over the course of your career armor gives you ten enhancements and a shield gives you ten more, up to nine of each can be devoted to other defenses than AC.
Armored spellcasting has been a staple of D&D clerics from the beginning, for two excellent metagame reasons: first, the character that heals other characters should be the last to fall, because once he falls no one can heal him. And, second, to make up for the fact that their spells are generally not directly offensive in nature.
Domain powers, though, were new to 3.5 and were, basically, a bribe to players to make clerics more attractive. Traditionally, the healer role is less "sexy" to players and those domain powers are the equivalent of two extra feats' worth of power, except these provide abilities feats can't and potentially cut across class lines. This always struck me as a little much, and it still does.