Boom got it right. Elements are essentially you're first job. At like level 100, you rebirth into a real job, like a Paladin or Assassin. Note, all elements can do each "function", some just do it better. Also, the different characters don't matter, you could be anything with any character except Maria and Sid. Those are like meant for support and offense respectively
My "view" on the elements
Water: Support element, has Curing abilities. Highest magical defense ability.
Fire: Offensive element, has highest attacking abilities. Has many debuffing abilities as well.
Earth: Defensive element. Has highest physical defense abilities.
Wind: "Aux" element. It's focused on Speed, but it has no pros. or cons. really. It's pretty much a secondary attacker like Fire, but faster.