Backing is betting on it to happen, laying is being the bookie for somebody else (i.e. betting against it happening).

So if something is 4/1 and you back it, you'll win £4 for each £1 you wager. If you lay it, you'll win £1 for each £4 wagered.

Betfair exchange is matching up people on both sides of the bet, with a margin of commission in between the back and lay values.

Back means you bet on something happening, lay means you bet against it happening.