In a nutshell:
Placing a back bet is the traditional form of gambling - you believe an event will happen and bet on it happening at odds you want to take.
How does it work?
If you're a backer on a Betfair market, it's the prices in the blue column that you're interested in. The prices in blue have been offered by the layers, while those in pink are prices requested by backers.
Let's say you're betting on Manchester City v Chelsea. If you've decided that it's City you want to be backing, just click on the price on City in the blue column, choose your stake and press on the button that will process the bet.
When submitting a bet offer, bets can be returned as unmatched, partially matched or matched in full. If you are taking the price shown on screen, you will be matched on the majority of occasions. Markets change continually though so don't assume you are matched - always look at your bet slip to confirm it.
If you're not fully satisfied with the price on offer, you can request a better one. Click on the same price in the blue column as before but make it bigger; for example, 2.6 rather than the 2.5 on offer and once again, select your stake. Then it's just a case of waiting to see whether a layer is prepared to match your bet at the bigger price.