One of my PC's without handle animal bought a psuedodragon to have as a companion. Is that legal? He does not have the handle animal skill, and the creature seems seems way over powered. He is also a rogue and bought one because they are in the ultimate equipment guide. There is no reference to it being a companion in the bestiary. How should I handle this?
An "animal companion" is a special ability that mostly druids and rangers have. It is irrelevant to this discussion as your rogue does not have it. See What feats can I use to get a companion of any sort? for a list of abilities that actually grant companions in the technical sense; these guys stick with you and help you because they're your nature buddy. The rogue, however, is basically in the same boat as someone who wanders down to the local zoo and convinces them to sell him a tiger. He has no specific control over it and it has no specific reason to care about him.
A PC can purchase and try to keep around any other kind of creature that's available for sale and they can afford, from animals to human slaves. Note that things are not available because "they are in a book the player has" - you're the GM, put on the big boy pants, you control the game world and what is available and how in the markets of your game world. If such a creature is available by sale or by capture, whether you can use them effectively and whether they kill you in your sleep or not has a lot to do with other rules systems and roleplaying.
If you have an animal (a monster of type "Animal" - not magical beasts, not anything else) you can use the Handle Animal skill to teach them specific tricks over a long period of time and to try to command them in combat. There are a couple magical beasts that have an exception stated to this (hippogriffs, familiars) that can have Handle Animal used on them in specific circumstances.
If you have anything else, then it's pretty much you convincing them through other means to help you out. This is one part GM fiat, one part bluff/intimidate/etc (for intelligent creatures). Charm or dominate, or the Leadership feat, or other specific rules in the game that allow for influencing other creatures apply as normal.
Pseudodragons have human-like intelligence; you do not use Handle Animal on them. You talk to them, and convince them to work with you; a typical pseudodragon understands Draconic, but can speak to any intelligent creature with its telepathy. Buying a pseudodragon effectively means the character bought a slave, and pseudodragons explicitly resent being treated as pets.
As smart as a typical humanoid, they do not enjoy being treated as pets and prefer being treated as friends. They are wary of evil folk but can bond with sorcerers and wizards as familiars, and some have befriended druids and rangers or partnered with good dragons as scouts. Pseudodragons will serve as familiars if they approve of a spellcaster's personality (and if the spellcaster takes the Improved Familiar feat), but often also bond with those whose company they enjoy or who have proven themselves true friends. A pseudodragon might follow another character in this manner for days, weeks, years, or even a lifetime if the creature is treated well, provided with food, and generally well-loved.
So the character can either free the pseudodragon and try to convince it to help out of friendship, or the character can go all-in on the slaveowner thing and use threats and punishment to demand obedience. Either way, the pseudodragon is a full NPC with his or her own identity, personality, plans, and desires, and should be roleplayed by the DM as such.
Also, for completeness, pseudodragons can be bonded to arcane spellcasters who have the Improved Familiar feat.
The DM Must Determine Why This Happens in His Campaign
According to this list a psuedodragon has a cost of 200 gp. Here are two ways to run this, but the DM is free to determine others as the cultural, logistical, and infrastructural requirements of the psuedodragon trade are campaign-dependent.
Pseudodragons are slaves. The rogue bought a slave--that possesses telepathy, poison, a fly speed, darkvision, a massive Stealth skill, and an average Intelligence score of 10 (smart enough to evaluate his owner), Wisdom score of 12 (clever enough to wait until he's asleep), and Charisma score of 10 (cute enough the warrant sympathy).
This is an incredibly dangerous creature, and by buying a slave the rogue has proven himself to be if not outright evil then at least okay with owning intelligent creatures, probably going against the pseudodragon's own Neutral Good alignment. I'd expect the pseudodragon to stop at nothing to escape... then possibly find the one who originally captured him and either get revenge, free other imprisoned pseudodragons, or both.
This jibes with part of the description given wherein prices are assigned:
Found primarily in temperate forests, pseudodragons are intelligent enough that some consider their sale slavery.
Pseudodragons are fundraisers. They allow themselves to be sold by a those of the appropriate alignment and bought by another of an appropriate alignment. That 200 gp goes to the church, orphanage, or whatever that raises pseudodragons, and the pseudodragons are pitched more as companions rather than combatants or hyperintelligent pets. The pseudodragon's telepathy allows it to interrogate potential buyers, and a pseudodragon is "sold" only upon the pseudodragon agreeing to the sale.
This also jibes with part of the description given wherein prices are assigned:
A handful of pseudodragons offer themselves for hire to adventurers, typically for half their listed price per week of work.
In such cases it might be better to say that the rogue hasn't so much purchased a pseudodragon as hired one to accompany the party for a few weeks.
As previously mentioned, one no more uses the skill Handle Animal on a pseudodragon than one does on a for-reals true dragon. The skill Handle Animal is for use on an animal or "on a creature with an Intelligence score of 1 or 2 that is not an animal...." A creature who wants to command a pseudodragon asks a pseudodragon; he doesn't handle the pseudodragon.
In Pathfinder's ancient antecedents pseudodragons were special. One only acquired a pseudodragon via the spell find familiar (and then only on a lucky roll) or via adventuring. The Monstrous Manual for Advanced Dungeons and Dragons, 2nd Edition says
The pseudodragon that searches for companionship will stalk a candidate silently for days, reading his thoughts via telepathy, judging his deeds to be good or evil. If the candidate is found to be good, the pseudodragon will present itself to the human as a traveling companion and observe the human's reaction. If the human seems overjoyed and promises to take very good care of it, the pseudodragon will accept. If not, it will fly away.
Pathfinder's shops full of pseudodragons--given their previous rarity--seem distressingly odd.
Under that specific circumstance, the answer would be no. Pseudodragons are intelligent enough as KRyan has said, and for a Rogue they wouldn't even be able to get it via the Improved Familiar feat.
They could however make it work with the Dragon Cohort feat found in the 3.5 Draconomicon on pg 104. If they insist on going the route they've gone and don't want to pop a feat on it though, I would highly suggest you play with the Pseudodragon as you see fit. Make that Rogue earn having the Pseudodragon around, or have it flee if it so wishes. If you're feeling nice, make it up to them with a shiny trinket later, but I wouldn't refund their whole gold investment unless they earn it somehow.