Most racials aren't super important, but they're nice bonuses.
For Priests it's a whole 'nother story because they have special spells for their races. Like how Dwarf has Fear Ward, Undead has Devouring Plague, Nelf has Starshards, etc.
But for other classes, racials don't particularly matter unless you're trying to min/max. The big ones are the weapon skill bonuses, which can be nice, and Undead's Will of the Forsaken, which is stupidly OP as it's essentially a 2nd PvP trinket against most classes.
If you're going to be an alliance healing priest, Dwarf or nothing. Fear Ward is really useful, and will get you raid spots for some fights like onyxia. Not to mention, Stoneform is super useful because it essentially allows you to be immune to all standard debuff types, except curses; being that you can dispel magic/disease on yourself, and use Stoneform to dispel poison/bleed. Really funny when a rogue tries to blind you to reset the fight, only for you to stoneform out of it and wreck them.
But really, all of the races have good racials. Human gets some extra spirit and rep and a stealth detection cooldown, and Nelf gets dodge chance and a nice stealth ability, which can be really nice for avoiding gankers.