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From FFXIclopedia, the free Final Fantasy XI encyclopedia

Each element has a weakness and a strength.

Fire > Ice > Wind > Earth > Thunder > Water > Fire ... and keeps going.

Additionally Dark <--> Light are opposed elements.

Or, if you like lists:

Fire melts Ice

Ice compacts Wind

Wind erodes Earth

Earth grounds Thunder

Thunder electrolyzes Water

Water douses Fire

Light illuminates Dark

Dark eclipses Light

Certain weapon skills have elemental properties, as well.

Ex: Red Lotus Blade > Fire

See also:
Weather
Ninja: Elemental Wheel
Calculating Magic Damage