You can search the GRO index for free if you know the surname someone was registered with at birth. The index will show their birth name, the quarter in which they were registered, the registration district and the mother's maiden name.
So, for example, if you know your grandmother was called Eileen Wilkinson and she was born in 1921, you can search the GRO index for her. The index should then tell you her mother's maiden name. Obviously this works better if the person you are searching for has a more unusual name. If you're looking for your grandfather David Jones from Swansea, it will be more difficult if not impossible to work out which one is him.
You can pay to get their birth certificates, it's about £6 for a PDF or £11 for a paper copy. But often you can get enough information just from the index to be able to get the surnames of the generation before the last one you know.
Again, if one or both of the surnames is unusual then you'll have better luck here.
So if from searching the index you're reasonably sure that your great grandparents were called Mr Robinson and Miss Brewer, you can search Ancestry to see if you can find a marriage record for them. If you can find out when they married, you can then search the GRO index for male and female children with the dad's surname and mum's maiden name, starting the year they married for up to 20 years to identify all the children they had together.
The GRO records go back to 1837, which means that once you get back past 1911 you can cross reference what you find against the censuses, which are all available on Ancestry.
Any further back than that and your best bet is parish records. If your ancestors were married or baptised in the Church of England in a big place like London, those records are often available.