In the World of Warcraft, each race has a small set of bonus traits that lend themselves to a greater or lesser degree of usefulness during game play. We're going to look at each race and the bonuses involved and draw a few conclusions.

Some bonuses lend themselves to certain classes, and others help in play style. Sometimes you just don't have a choice. Like if you want to be a horde druid – you're getting a Tauren and that's all there is to it. But in the cases where you do have a choice, it can actually make a fair amount of difference in your character in the earlier parts of the game.
Racial bonuses have a tendency to balance out for the most part by level 70. You're still going to see mostly Tauren Main Tanks, but most races can do a great job of playing the classes available to them.
As well, certain classes match up well with certain racial bonuses. Undead have the highest base spirit in the game, just like night elves have the highest agility. That's why you see so many night elf Rogues. Every race but the Dranei can make Rogues, but you will see more night elf and gnome Rogues than human and dwarf. The bonuses to agility for night elves and the escape artist bonus for gnomes lend itself to playing a Rogue. That's also why you see so many gnome casters – gnomes get a boost to intellect higher than any other race.
When it comes to try out a shiny new toon, let's first look at the useful, and sometimes not so useful racial attributes for the different races and how they might match your particular play style preferences.
We at first talk about Humans race, Humans have one racial trait that makes them great Priests, increased spirit, the rest are just OK.
On the one hand, Humans have increased stealth detection. The vast majority of WoW players consider this completely worthless, because it basically is.
On the other hand, Humans have higher stats for spirit than the other alliance races, therefore a lot of human priests and Humans get an extra 25% reputation when turning in quests–cool but it still won't get you exalted with gnome Regan that much faster so you can ride the giant turbo chicken mounts.
Dwarves don't seem to have a single bonus that's truly useful, unless you consider being a Paladin a bonus. For a short period of time, stone form greatly increases defense. It's OK, but won't save your life very often.
For a Hunter, +5 gun skill is not bad if you don't mind the loss of agility from being a dwarf in the first place and frost resistance is not a big enough bonus to count for much in PVP and very few mobs use frost attacks in PVE.
Night elves may have the second-best racial bonuses on the Alliance side, but they are still mostly best for Hunters and Rogues. Shadow meld is a bit like stealth, only you can't move. If you stealth and shadow meld you are virtually undetectable. Dodge bonus are very helpful for Hunters and Rogues who have attacks that come after dodges and a good bit of their defense rating is based on not getting hit in the first place.
For one thing, Gnomes come in behind night elves for helpful bonuses, but I just can't get used to having to swim where others walk; -10 points for pure shortness. For another Dranei really doesn't have a completely useless bonus, even though the resistance to shadow is small.
This is all about the racial bonuses and does not take into account the stats each race starts with.