When you buy a house that you can't afford to buy outright (which is the case for the vast majority of people), you take out a secured loan (in this case, a mortgage) in order to buy it and agree a rate of interest and a schedule of repayments with the lender.
The loan is secured against the house. In other words, if you don't, or can't, keep to the repayment schedule the lender can then recover the full amount of the loan from you by forcing the sale of the house through a repossession order.
The house belongs 100% to you. But there's a loan secured against it, so you must keep up with repayments.
So, now onto what we call "equity". The difference between what the house is worth and what you still owe to the mortgage lender is your equity.
When buying a house you normally pay a deposit up front and the remaining cost of the purchase is from a mortgage loan, so the amount of deposit you paid will be your equity initially.
Over time, the value of the house may go up but your mortgage amount remains the same, so the amount of equity you have increases.
Here's an example. You buy a house in Year 1 for £250k, paying a deposit of £50k and taking out a mortgage for the remainder. So, at this point your equity is approximately £50k.
Over time, the value of the house may increase. Perhaps by Year 10 it will have increased to £400k, for example. In that case, your equity would be £200k if you had an interest-only mortgage or somewhat more if you had been repaying capital on the loan. (ie equity is the value of the house minus what you owe).
When you sell the house, providing the house is your main home and isn't a 2nd home, holiday let or rental property, then all of the equity remaining after the sale has completed is yours and there is no tax to pay.
If the house is not your main home then you would need to pay capital gains tax on the amount that the property had increased in value.