The Player's Handbook describes eight player races: Dragonborn, Dwarf, Eladrin, Elf, Half-Elf, Halfling, Human, Tiefling. The Player's Handbook II describes five more: Deva, Gnome, Goliath, Half-Orc, Shifter. Monster Manual I lists 15 more candidates for player races: Bugbear, Doppelganger, Drow Elf, Githyanki, Githzerai, Gnoll, Goblin, Hobgoblin, Kobold, Minotaur, Orc, Shadar-Kai, Shifter (Longtooth), Shifter (Razorclaw), Warforged. If you look through other Wizards of the Coast material, on forums, through 3rd party sources, etc, you will find many more races that some people consider acceptable player races.
What Should You Allow?
You should allow whatever works for you and your players. If they want to play monsters, the rules offer support to make that work. If they want to play a small talking dog, there is no reason that they can't if you allow it. However, non-traditional player races will mean a great deal more work for the DM. Monster tend to be evil. Or at the very least, they aren't welcome in most civilized areas. If these characters can't go into town, where do they get their provisions? If all the players create the same type of monster then maybe you could make a community of that sort of monster for them to use as a home base. It is doable but you are stretching some of the rules pretty far to cover these races as player characters. Give some thought to whether your world can support non-traditional player races and what changes you'll have to make for them work.
My Personal Opinion
Your opinion is always more important than mine, or anyone else's, when it comes to selecting house rules. Some are hard to implement but it is ultimately your choice as to how to run your game.
My opinion is that no player is at all capable of running a character that is any race other than Human. What you end up with is a bunch of players running Human characters dressed up as Dwarves, and Elves, etc. Players don't know what it is to be a Dwarf so how are they expected to run one? They can read the definition of a Dwarf till they are blue in the face, but they can never fully understand what it is to be a different race.
Many guys run female characters. Many think they are doing a good job. Ask a woman her opinion of how these guys are doing. Some may actually be doing a fair job but the vast majority don't have a clue. And it is a far cry from a guy playing a girl to a human playing any other race.
Beyond the question of whether someone can play another race is the consideration of bringing that race's knowledge out into the mainstream.
In my world, Elves are immortal. Dwarves and Gnomes live many hundreds of years. Even Hobbits live considerably longer than Humans. If a player can make a character who is an Elf, he may be fairly young himself but he has grown up around immortal beings and has been exposed to knowledge of the workings of the world that are many thousands of years old. This results in a character with potentially sage-like knowledge of the world. That makes it very hard to have secrets.
Again, in my world, the old races are close-mouthed about much of the history of the world. Knowledge is power and they like to keep that power for themselves. I have spent a great deal of time developing ancient races that no longer exist that have left behind artifacts and ruins that I don't want players to know anything about it. When they find any hint of these people, it is a great secret that they are seeing just a hint of. It's hard to rationalize their ignorance when there is a character in their midst who grew up around people who personally knew this ancient race.
Furthermore, some of the races in my world have abilities and skills that aren't shared with outsiders. Players will never know about them but I use these powers in building my back-story to support how certain events transpire and where some items come from. These are secret things that players shouldn't know.
Does That Mean That Players Can Only Make Human Characters?
I actually tried that once. Characters had to be Human but they could hire non-human hirelings. It didn't work out. No one was happy, including me. I scrapped that idea very quickly.
Now, I allow players to make characters who are: Human, Dwarven, Elven, Gnomish, and Halfling. However, none of the races are Dwarf, Elf, Gnome or Hobbit!
Dwarves and Gnomes live deep in the mountains. Elves have secretive homes in the forest. Hobbits live in secluded pastoral settings. Each of these races are isolationists and have little to do with anyone but themselves. Over the years there have been romances between Humans and these other races. And by some wonderful chance, these couplings have been able to produce offspring (Sometimes you need to throw realism out the window). These offspring are shunned by their non-human relatives so they often end up living in Human societies.
As a result, most people in towns think they are living amoungst Dwarves, Gnomes, Elves, and Hobbits. Instead, they are living amoungst Half-Dwarves, Half-Gnomes, Half-Elves, and Half-Hobbits (or Halflings). Even those that know the difference find it cumbersome (if not insulting) to say Half-Dwarf so instead they are just referred to as Dwarves.
This allows me to have long-lived (and even immortal) races that maintain their racial knowledge and secrets while allowing players access to the traditional non-human races. These "half-races" still match the descriptions listed in the Player's Handbook. The "true-races" are far more different and require write-ups to replace those in the Monster Manual.
So What About The Other Races?
Personally, I don't allow any other races. They just don't fit with my vision of the world that I have created. I don't allow monster races (or evil characters for that matter) as I don't want to deal with the problems faced by characters who can't enter normal towns and villages without conflict. The other races may not even exist in my world. I'm careful about placing humanoid and monster populations to fit what I want to accomplish. Some of the races simply haven't been added to my world which makes it impossible for there to be characters of their races.