There are many character archetypes that have animal companions: druids, hunters, rangers and mages, for example. Since we've been talking about eagles here are some things to keep in mind when allowing a player to have an eagle as a companion.
1) An eagle has a wing span of 6 feet +. Yes, bigger then a person might be tall. Cool, huh? Though they are a bird, they are still quite sizable and have wicked talons. (Though in most monster books normal eagles do very little damage, their talons do as much damage, in my opinion, as a dagger or even a short sword they just don't have the reach.)
2) Allow the eagle to grow. In Pathfinder, the druid's pet does actually grow as the druid advances in levels. This allows for more damage, and if the character happens to be a gnome, they might become large enough to carry their gnome friend. Even if your campaign system doesn't allow for growth, in either size or power, adjust. This allows the character to have a cool companion that they won't want to swap out as soon as that cooler, stronger monster comes along.
3) Remind your character's player that eagles, and other flying creatures, do not necessarily like enclosed areas, making dungeon runs hard. People tend to forget birds do not like caves and such and get mad at their DM for saying their companion doesn't want to follow. If you remind them up front, they have no leg to stand on.
4) Remind the player to treat the animal right. If the player forgets about their pets, except during a battle to save their but, perhaps the pet should suddenly not be there...