Aside from Artisans' Tools, what other equipment do I need to use a Craft skill? More specifically, the Craft Armor and Craft Weapons skills. I bought a hammer as I can also use it as a weapon, but do I need to buy an anvil as well? And if so, what size? And, if an anvil is definitely needed, would it be cheaper in the long run to rent one rather than buy one?
Before committing to this route, ask the GM if the campaign will even accommodate a PC using during play the skill Craft. Many campaigns don't or can't accommodate mundane crafting well at all because mundane crafting takes a long time—a basic sword can take weeks and an expensive, extravagant item made of special materials can take from several months to over a year of constant work. Even a master armorer or weaponsmith often finds it more convenient to purchase his gear instead of making his own. (The Craft skill can be used untrained, though, so it's totally legit to compose a character background saying that a character was, like, a weaponsmith's apprentice before he started adventuring yet the character possesses no ranks in Craft (weapons), quitting his master before he took any ranks. That's a thing.)
Tools and raw materials make crafting happen
Assuming the GM's campaign supports mundane crafting, as other answers mention, at least improvised tools if not artisan's tools or better are needed to practice a craft. These are abstracted sets of generic tools needed by a craftsman to practice the craft. Unless the GM rules otherwise, these are all the tools needed to practice the Craft skill. A separate anvil, for instance, shouldn't be necessary unless hunting roadrunners or needed for a feat. Also, usually tools and weapons are separate things—a warhammer suitable for bashing in orc skulls may not be a good tool for hammering steel.
Keep in mind that each set of artisan's tools is good for but one Craft skill—artisan's tools for the Craft (armor) skill are not also applicable to the Craft (weapons) skill, for example. Luckily, Craft skills are broad—the the Craft (armor) skills covers all armor and shields from studded leather to heavy wooden shields to full plate and the Craft (weapons) skill covers melee weapons from quarterstaffs to flails to greatswords—so the tools are broad, too. To create a quarterstaff a craftsman does not need artisan tools for both Craft (carpentry) and Craft (weapons), for instance.
Extrapolating from existing rules: Additional crafting requirements
A creator that's creating magic armor or a magic weapon needs "a heat source and some iron, wood, or leatherworking tools." It's not unreasonable for the GM to mandate that a craftsman that's crafting mundane armor or a mundane weapon meet the same requirements.
This GM views the required tools as part of an appropriate artisan's kit, not requiring, instead or in addition to artisan tools for the skill Craft (armor) a separate new and different set of artisan's tools for the skill Craft (leather goods) if creating or crafting magical or mundane leather armor, for example.
However, this GM also view the necessary heat source as needing to be significant—attempting to blacksmith up a greatsword over a flickering candle is a futile endeavor, for example. Fortunately, in most places where a craftsman plans to craft, finding a roaring fire shouldn't be a big deal. (In a pinch, I'm certain another PC'll arson something up for you.)
Finally, a craftsman usually can't make something that costs something from nothing, so a craftsman typically also needs raw materials. These are the abstracted parts that are needed to create the item, and these raw materials usually cost 1/3 the cost of the finished item. For instance, a longsword costs 15 gp, so the raw materials for a longsword cost 5 gp. Ask the GM if a craftsman buys raw materials that are either generic and can be used to craft whatever or broad and can be used to craft anything within a certain skill or specific and raw materials must be purchased for each item the craftsman intends to create. With the looming potential of an accounting headache, pray it's not the last.
You actually only need improvised tools. Though those apply a -2 penalty to your checks, so you want at least Artisan Tools to remove this penalty.
Other than that, Masterwork Artisan Tools will grant a circumstance bonus +2 bonus, getting help from assistants will grant another (untyped) +2 bonus on your check. The limit on the number of assistants is defined by the circumstance and your GM.
Using the (optional) downtime system from Ultimate Campaign, the Artisan Workshop, the Workstation and the Forge all function as masterwork artisan tools, but each of them for up to three different people, though a forge is limited to metalworking skills such as Craft(Armor) or Craft(Weapons).
Which means that Artisan Tools are all you need if you are working by yourself. All necessary tools are assumed to be among those tools.
If you require an extra bonus, a potion of Crafter's Fortune (or making friends with someone who knows the spell) will grant a +5 luck bonus. There are magic items that will grant you bonuses as well.
By the way, there is a handy crafting calculator for Pathfinder.
Personally speaking, I think this goes more along with how much your DM would like to delve into the details. For my game, the DM lets us craft so long as we meet the monetary and skill requirements. Here's an example:
My dwarf wants to craft an Armored Coat. The craft DC on armor is 10 + the armor bonus, so in this case, 14. In order to find out how much it would cost, we took the price, converted it to Silver (50 gp -> 500 sp), and the "materials" cost 1/3 of that. That comes out to 166.6666666..., so we just rounded up to 170 SP. In order to make progress on completing it, I have to roll a Craft(Armor) check and beat 14.
In that case, the anvil, necessary metals, tools, etc. were all covered by that price that was paid. That isn't to say that having tools doesn't help because Masterwork Artisan tools give you a +2 on your craft roll. I can clear any of this up if you'd like, but I pulled the information on crafting from here: http://www.d20pfsrd.com/skills/craft/