A spec is what talents you have mainly oriented on. A fire-mage for example has a fire spec.
You can choose any talents you like in your talents once you have reached level 10. It is advised that you go deep in one 'tree' before grinding talents off an other.
Again as an exemple, a fire mage will go deep into a fire tree so he gains many benifits out of using fire spells. Very rarely will he be using frost or arcane spells as his fire spells will have great improvements thanks to his spec.
You can go deep into any spec, it's all question of what you feel comfortable with. If you are leveling up I'd suggest either fire or frost, fire = lots of damage but very vulnerable to attacks, frost = great control and pretty sweet crit combos and mana efficient.