From Which... but your options are significantly limited.
Second-hand car bought privately
You have fewer rights when you buy a used car from a private seller, and key parts of the Consumer Rights Act don't apply.*For example, there is no legal requirement for a car to be of satisfactory quality or fit for purpose.
But contractual rules about misrepresentation do apply. So, legally, the seller must:
accurately describe the second-hand car. For example, an ad must not say 'one owner' when the car has had several, not misrepresent the second-hand car, ie tell you something about it which isn't true. For example, if it's been in an accident, the owner mustn't tell you it hasn't.
If you're buying from a private seller, the onus is on you as the buyer to ask all the right questions before making the purchase. The seller doesn't have to volunteer extra information so, if you don't ask questions, you may not have the full picture of the car's history or be aware of any potential faults.