Contact your local council
The best place to start your search is to contact your local council. Most councils will have a contact (usually in their parks department) that will be responsible for all the allotment sites in the Borough. I highly recommend that you phone your council if possible rather than write to them as you will then develop a rapport with the contact, and find that they are far more helpful than if you simply write or send an email.
There is a great tool on the government website that will give you the contact details for your local council.
Your local council will be able to tell you where your nearest site(s) is, the cost of renting an allotment, and whether there is a waiting list (and how long). It is estimated that there are 100,000 people on waiting lists for allotments across the UK, and in some boroughs they can be as long as 10 years. Don't let this put you off however! If you want an allotment and there is a waiting list, then sign up immediately (you can always look for other options while your name gradually makes it's way to the front of the queue).
The anecdotal evidence suggests that even if you are quoted a waiting list of 10 years or more, you are likely to get a plot much sooner than this. Many people on the current waiting are also on lists elsewhere, and by the time they make the front of the queue they already have a plot and some on the list will have changed their mind.
Visit individual sites
OK, so your on the list, and waiting for your plot. The next thing you can do is go and visit the sites you are interested in. Talk to the other plotholders there. You may find that someone is looking for help with their plot and you could share their plot, or even better you may find that they will have local knowledge of plots that will become vacant in the near future, and if the list is not too long, then this knowledge could prove useful in your discussions with the local council.
Look further afield
If you have a car, or bike, then you may wish to look further afield for your plot. Local councils do not hold plots for residents of the borough, and you may find that the waiting list is much shorter if you are willing to travel a few miles.
If you go down this route then make sure you check to see how long it will take you to get there, and whether transport links are sufficient. A plot that is 45 mins away from home is not going to be visited and tended as much as one that is 10 mins away... and if you don't look after your plot then the council will take it off you and offer it elsewhere.
Failing all this have a look at the Landshare Scheme. This was launched by Hugh Fearnley-Whittingstall on the Channel 4's River Cottage program a few years ago, and is an ingenious solution to the problem of lack of allotment space. It connects people with spare land, with people who are looking for growing space... and the only cost is a healthy share of the produce grown on the land.