The current you can put through it is called "carrying current", the current you can switch is "switching current" or some other similar term such as "load current" or "contact current". For example:
Note that you are only allowed to switch 7A @ 24VDC or 10A at 110VAC with stated inductive load types, but you can carry the full 15A.
Most relay datasheets do not have specifications that permit you to exceed the resistive switching current, in my experience, so if you did try it you'd be exceeding the maximum ratings.
If you use a 20A relay and your switching current is much less (and resistive) you'll get much longer life than if you switched 20A.