Pay off your debts
If you have any debts, not including your mortgage (although you might consider paying a chunk off that if you're allowed), you should use the money to get rid of them for good before you do anything else.
Cash in on ISAs
Firstly, you might get an instant access cash ISA, which is basically a tax-free savings account. This means that if the interest rate is 3%, you get 3%, not just the 1 or 2% you'd get in a similar, boring savings account. You can put up to £15,000 in a Cash ISA every year. But you could put £5,000 in this year.
Stash in a savings or current account
Next, use a good old instant access savings account or current account. Yes it's taxed, but you can earn more than by just leaving the money under the mattress. Let's say you put £5,000 in a savings or current account, so you now have £10,000 in savings.
That's a great safety net of cash, which you can get at in a hurry if need be. Plus, as long as you keep moving your funds to the accounts with the best interest rates, you should stay ahead of rising prices (inflation).
Save some in a stocks & shares ISA
However, simply leaving this money in cash is arguably inefficient and wasteful. You should consider how to make this money make more money.
Your first step might be to open a stocks & shares ISA. You can put up to £15,000 into one of these each tax year (although you can only put £10,000 in if you also opened a Cash ISA in the same tax year). Stocks & shares ISAs are free of Capital Gains Tax, which means it doesn't matter how much the shares rise, you won't pay tax on it.
It's best to start investing with a lower-risk investment, such as an index-tracking ISA. These things track indices (it's all in the name) such as the FTSE 100 or the FTSE All-Share. As share dealing in index trackers is automated, they are usually very cheap. Plus they often outperform funds managed by humans! If you can afford to add to them later, it's a good idea to dripfeed money in over the year.
Put some in a pension
If you don't need the money now, you could put the rest of the money into a pension. The great news is you can get index-tracking pensions too! Your first step should be to look at the funds offered by your company, as they probably match all or part of your contributions. If no such scheme is available, you should look online for index-tracking stakeholder pensions that have low total expense ratios, which are the administration costs. Less than 1% is good.
However, remember that investing is a long-term endeavour. With index trackers you can usually expect your shares to do very well thank you over five-plus years, but all markets fluctuate. If you're saving up for something, or if you need the money in less than five or so years, investing it probably isn't a good idea. If you need the money before retirement, then investing it in a pension isn't a good idea at all!