A Back Bet is a bet on something to happen - This could be a bet on a team, player or horse for example. You would only win the bet if your bet selection wins.
A Lay Bet is a bet on something NOT to happen - This could also be on a bet on a team, player or horse for example. You would only win the lay bet if your bet selection did not win.
Back Bet and Lay Bet Football Team Example:
Match: Liverpool v Chelsea
A Back Bet on Liverpool to win would only be settled as a winner if Liverpool win the match.
A Lay Bet on Liverpool to win is a bet on Liverpool to NOT win the match. The Lay Bet would only be settled as a winner if Chelsea win or if the game ends in a draw (so two outcomes are playing in your favour).
Back Bet and Lay Bet Horse Racing Example:
A Back Bet on a 'Horse A' to win will only win if 'Horse A' finishes in 1st place and wins the race.
A Lay Bet on 'Horse A' to win will only win if any other horse in the race finishes in 1st place (so all the other horses in the same race are playing in your favour).