Pets have five personality traits, which can be adjusted before creating or buying them. If buying a pet from the Adoption Centre, you will not be able to change their personality, but it will be displayed. Different personalities affect how your pets behave, or how well they learn commands.
Genius pets will learn commands quicker than a doofus pet. This is very beneficial if your Sims have little time on their hands.
Pets that are hyper will dash around everywhere. They will howl or yowl more often too, even if you fill the Calm metre to the top. Lazy pets lead a quieter life, and will spend more time snoozing than demanding games.
Pets that are independant like to do their own thing, and so their Social need will take longer to decay. However, this won't help your Sim if they are living by themselves. Pick a friendly pet and they'll ask to be stroked or have a tummy rub, and won't want to leave your Sims alone!
Pets that are Aggressive will be far more hostile towards others, growling and hissing and picking fights. If your Sim wants a bodyguard, this could be an option. Cowardly pets will simply stand defenslessly if challenged, but won't try to growl of hiss at their owners.
Pigpen pets will happily roll in dirt or eat garbage, which may be inconvinient for your Sims if a smelly dog comes in the house. If a pet if Finicky, however, they'll only want to eat from a clean pet bowl, and cats won't use their litter trays if they're full of gunk.