Savings accounts are not dissimilar to Current Accounts, but in exchange for a higher rate of interest you are more restricted in what you can do.
Savings Accounts encourage you to keep your money in longer, and should generally be used for storing cash you have no immediate use for and won't need access to quickly. The terms and conditions of accounts will vary, but in general there will be a limit to how much you can withdraw in any given period, and there may be a notice-period before you can get access to your cash (though this practice is in decline).
Savings Accounts won't have any mechanism for paying bills - you will have to transfer cash to a current account to do that. You may be able to withdraw cash from a cash machine with this kind of account. You will not generally be given an overdraft facility.
You will receive a higher rate of interest for cash kept in this kind of account. It is still a low-risk account however, as you will always get back at least as much as you put in.
Most banks or building societies offer savings accounts. The actual terms and conditions may vary, so it is worth shopping around. Bear in mind that the terms and conditions of your account may change over time. Certainly interest rates will change as base rates fluctuate.
You will be required to prove your identity (by law, as part of measures against money laundering) before being allowed to open a Savings Account.