For interest earned from, for example, bank accounts, savings accounts and government bonds (full list is here), you pay tax on any interest earned over the Personal Savings Allowance (PSA).  

If you earn more than £17,750, to calculate your PSA, add all the interest you have earned onto all your other income.  Of note, for the PSA calculations, the rest of the UK tax rates are used, not the Scotland rates:

• If the total brings you into the basic rate, your PSA is £1,000.  For anything over £1,000, you pay tax at the rate you pay on your other income.
  
• If the total brings you into the higher rate, your PSA is £500.  For anything over £500, you pay tax at the rate you pay on your other income.
  
• If the total brings you into the additional rate , you do not get a PSA and you pay tax on all of your interest.

If you earn less than £17,750:

• You are eligible for the starting rate for savings.
• Your starting rate for savings is a maximum of £5,000.
• For every £1 of income above your personal allowance (£12,570), the starting rate for savings is reduced by £1.
• For example, you earn £15,000 per year and have £2,000 in interest from savings.  Your starting rate is £2,570 (maximum starting rate (£5,000) minus £1 for every £1 you earn over £12,570 (£2,430).  You therefore do not need to pay any tax on the interest you have earned. 
• Lastly, if you have any Personal Allowance remaining, you can use it to earn interest tax free.

Assuming you are in employment, HM Revenue and Customs (HMRC) will automatically take your interest on savings owed by adjusting your tax code (your bank should report the interest you earned to HMRC).  If you're self-employed or earn over £10,000 from savings and investments, you will need to include details of the interest earned in your Self Assessment tax return.

If you believe you have overpaid tax on interest, you have 4 years to claim from the end of the relevant tax year.  You can reclaim through a Self Assessment tax return if you complete one, or by using Form R40.