So basically every character has an attribute, which enhances their resistance against that attribute. The magnitude of the resistance is based on the character's tier. Here is the list:

Tier Increased Resistance Against Their Element
1 +10%
2 +8%
3 +5%
4 +5%

Also, the resistance gets doubled when the character reaches +5 evolution.

Then there are devil fruit characters, which have -15% water resistance, therefore, water based characters have this slightly compensated with their increased resistance. For example, Aokiji has -7% water resistance instead of -15%. (-15% from his devil fruit + 8% from his water attribute resistance = -7%) Once evolved to +5 he should have +1% water resistance.

This is how characters reduce resistance with basic attacks and skills (The debuffs of basic attacks and skills are considered different, therefore the are able to stack) :

Water reduces Fire reduces Light reduces Dark reduces Thunder reduces Water.

Elemental chart

As you have noticed, characters reduce enemies' elemental resistance and you can only reduce one resistance (they don't stack) at a time. In case two elemental debuffs are acted upon the same character, the latest one overrides the previous one. Basic attacks and skills are considered different, therefore if both are done consecutively, then the Resistance Reduction will stack.

I am not certain of how much resistance those debuffs reduce, but I am quite sure they are pretty noticeable. So taking this in consideration when building up your team will maximize your damage output and defense :). (For example, using Shanks behind Mihawk will increase Shanks's damage if Mihawk has greater speed, since Mihawk reduces thunder resistance) Also, don't confuse Enel's Resistance Reduction with the elemental debuff; the Resistance Reduction is a skill, which is considered as another debuff, so you can add up to 3 defense debuffs on the same character.

You can check any characters' attributes by clicking on their stats in the advanced talents window.

Screenshot 2015-01-18-20-10-44-1-

Attributes window

I think this covers all the elemental information around the game. Let me know if I have forgotten anything.