Note: the following is a very rough first draft to be refined later.
Roughly speaking, betting odds imply a probability that is the inverse of the (decimal) odds. If this implied probability is lower than the probability perceived by the bettor, the bettor may want to back. Otherwise, the bettor may want to lay.
For example, consider tomorrow's Real Madrid versus Liverpool. At the moment, on Betfair,
Decimal odds of 2 imply a probability of Liverpool winning of ½ =50%. If you believe that Liverpool's probability of winning is 60%, higher than the implied probability, then you may want to back Liverpool, for the expected profit is 20 pence for each pound you bet. Bettors on the other side of the bet may believe that Liverpool's probability of winning is lower than 50%.
You can be a bet-maker or a bet-taker. If you offer odds of Liverpool winning of 2.1 and are willing to bet 1000 pounds, you only need bet-takers willing to back Liverpool to the tune of 1000 / (2.1 - 1) = 909 pounds to have your order fully filled. If Liverpool wins, you lose 1000 pounds. If Liverpool loses or draws, you get your 1000 pounds back plus the 909 pounds bet by the backers. Note that 1909 / 909 = 2.1.
For the sake of simplicity, I am assuming that Betfair charges no fees. In reality, Betfair would take a bite of the winning bettor's profits.