This is the money the government (HMRC) will put into your pension to encourage and incentivise you to contribute more. It can be called 'tax relief' as it is not really the government's money they are handing out, they are essentially giving some of your tax back.
For example, if you are a ‘basic rate tax payer’ (you earn less than £50,000), you’ll being paying 20% of anything you earn above £12,500 straight to the taxman. In that case, say one month you earn £1,000 – you pay 20% tax (£200), so are left with £800. If you then put that £800 into a pension, you’d get the £200 back as tax relief from the government (straight into your pension pot).
There is a minimum amount everyone is entitled to, no matter what they earn. If you earn less than £3,600, you can pay in £2,880 into a pension and get £720 added by the government. However, you generally can’t get more tax added back on any contributions above your yearly income than you actually paid on your earnings. So if you earn£10,000 salary pear tax year, you could only pay £8,000 into a pension and get £2,000 tax relief, as this would equal your salary £8,000 +£2,000 = £10,000. I this example any pension amounts above this amount would not be eligible for tax relief.
However, if your income is above £60,000, and you wish to contribute above this amount you may have to pay a tax charge, unless you have unused pension contribution allowances which you may carry forward up to the previous 3 tax years.
If you earn above £210,000 per tax year or have taken pension income from your pension you may have a lower pension allowance, which means the amount you may contribute into a pension before any tax charges apply may be lower than £60,000.
Remember that the amount you can get in total from tax top ups is dependant on your individual situation and can change in the future.
Find out more on MoneyHelper's article about the annual allowance for pension savings.