CONCORD, which is short for Consolidated Cooperation and Relations Command, is an organization that attempts to keep the peace through various means.
Increasing CONCORD standing is the same as increasing your security standing. You can increase your security standing is by killing NPC pirates.
No. At present, it is not possible to do missions for CONCORD. Your CONCORD standing is the same as your security standing, though, so if you kill NPC pirates and leave non-criminal players alone, you will eventually get a good standing with CONCORD.
CONCORD customs enforces trade restrictions in space. If your ship is carrying contraband (items deemed illegal in the current system), a customs vessel might warp warp scramble and stasis webify you, then contact you and ask you to hand over the illegal goods. If this happens, just click Yes in the message window to give the goods to the customs official. If you try to deal with the situation any other way when you are prompted to relinquish the contraband, the customs official will start shooting at you, and will quite possibly destroy your ship.
If you are carrying contraband, you will get a warning before entering a system where it is illegal.
In systems with 0.5 security and above, CONCORD will have ships flying around gates and NPC stations. CONCORD will come destroy your ship if you attack a player without provocation, or if you steal from a cargo container that isn't yours.