The line of succession remained more or less unchanged in its rules from 1701 when the Act of Settlement was passed, to 2013 when some of these were changed.
The Act of Settlement was originally passed to establish the rules for inheriting the throne to avoid a crisis that might be caused by the lack of an obvious heir. At that time the reigning monarch was Queen Anne, the last of the Stuarts. All her children had died in infancy, and her sister Mary, who had previously ruled along with her husband James, had had no children either; and by now it was clear that Anne would not be able to produce an heir to the throne (she eventually died in 1714, having had no surviving children after 17 pregnancies). Most of her other Stuart relatives, who could have had a claim to the throne, were Roman Catholics, who were barred from inheriting it since the days of Anne’s father (, James II, who was the last Catholic monarch) and would remain so for over 300 years. So the Act of Settlement was passed to ensure that the throne would pass, on Anne’s death, to Sophia of Hanover or her heirs. Sophia was related to Anne, being a granddaughter of James VI of Scotland and I of England, who was Anne’s great-uncle, and was the only non-Catholic relative who could be found. However, in the end she died a short time before Anne herself, and so the throne was inherited by her son, who became George I and despite speaking no English, was the official ruler of Britain until he died in 1727. All UK monarchs since then have been members of the House of Hanover.
In 2013 the line of succession was changed. Previously the Act of Settlement that the throne must always pass through male heirs before going to any females (a rule known as male primogeniture). However, before the birth of Prince George in 2013 this was amended so that the new baby would still become heir to the throne whether it was a boy or a girl. At the same time the law was changed so that the monarch could marry a Catholic.
This didn’t affect the previous line of succession, only any children born after 2013; so the old line which means, for example, that Princess Anne (Queen Elizabeth’s second child) can only inherit the throne after both her younger brothers and any of their direct heirs still stands.
The full line of succession goes back over 100 names, and so it is too long to give here. But in the immediate line to the throne we can see that the heir to the throne, ie the first in line, is Prince Charles, now the Prince of Wales but who in due course should become Charles III. Second in line is his elder son, William, now Duke of Cambridge. He can expect to become first Prince of Wales and then William V. Then William’s son George is third in line – eventually he would become George VII. Any children he had would then become next in line after him. If William has another child, that child will be next in line after George; at present though, the fourth in line is Prince Henry (usually called Harry).
After this we see how the old rules still operate for heirs born before 2013, because the next in line (number five) is the Queen’s third child, her son Andrew, the Duke of York. His two daughters, Beatrice and Eugenie, are sixth and seventh, then Edward (Earl of Wessex) is eighth followed by his two children James and Louise, and Princess Anne is 11th.
You can see the full line of succession here.