1) Government set a limit on the total amount of pollution that is permitted to be emitted into the atmosphere. In terms of cap-and-trade in regards to climate change, that's carbon dioxide. That's the cap. Emissions above that level result in fines.
2) Permits then get distributed to industry for a given amount of pollution. They can either be auctioned off (better, but polluters don't so much like this), given away for free based on historic levels of pollution (seemingly rewarding polluting industries) , or some combination of the those.
The idea is to set the initial allotment slightly below what's already being emitted -- the whole point of this is to reduce the amount of pollution, not just keep it at current levels.
3) If a company can reduce its pollution below the amount of credits it already has, it can sell those credits to some other company that's not doing so well on the reduction front.
4) This system essential creates an economic market where polluters have financial incentive to reduce pollution in the form of spare credits which can be sold or traded.
5) Over time the cap can be reduced -- hopefully at predictable intervals so everyone can prepare for it -- further constraining the amount of pollution that industry can emit.
Any more questions?