Loyalty cards are a shop's way of rewarding you for spending money with them. It's usually a points-based system, whereby you get so many points for each pound you spend in that shop. Some are good (Nectar (i.e. Sainsbury's, B.P. etc,)) and some are pathetic (WHSmith ClubCard - I've been saving for 156 years and almost got enough for a standard pencil (no rubber on the end though)).
Some would argue, however, that they are a false economy. They can tempt you to spend money in that shop when in fact the product you're wanting to buy could well be cheaper somewhere else. And the points you're rewarded with doing compensate for the extra money you're paying them for the product.
So they're only any good if you're going to use them in a shop where you're going to buy things anyway. For instance if you always shop in Tesco's and rarely anywhere else, then it may be an idea to get a Tesco Clubcard.
Make sure, though, that you don't inadvertantly sign up for a store card. These are a different thing entirely - they're like a credit card for that particular shop. So if you are tempted to sign up for something, make sure you ask them if it's a loyalty card or a store card.