As others have stated, there were two monarchs that ruled during Shakespeare's life, which was roughly from 1564-1616. I will try to add new insight into these two characters, as opposed to repeating what other answers have already stated.
1. Elizabeth the First (reigned from 1558 to 1603)
Elizabeth the First is one of the most famous rulers in the history of England. It was uncertain that she was going to rule, and many English people (particularly Roman Catholics) found Elizabeth to be an illegitimate ruler. She is famous for many things, including her extreme intelligence and cunning wit. Elizabeth the First is said to have known six languages. Shakespeare's early plays have many linguistic jokes, and this could be traced to his affection for Elizabeth the First.
2. James the First (reigned from 1603 to 1625)
James the First was initially the king of Scotland, but he also became the king of England in 1603. This unification created the kingdom of Great Britain, which exists today. One of his greatest contributions as king was the creation of the Authorized King Jame's Version of the Bible, created in 1611. This has continued to be the standard bible text today. James was a peaceful king, and often tried to unify countries. Shakespeare's enthusiasm with other parts of Great Britain could be attributed to the rule of James the First.
The first and longest ruling monarch was Elizabeth I who ruled for what might be thought of as the first half of Shakespeare's writing career. Many of the issues to do with marriage and the issues to do with whether one should marry according to the will of others was particularly pertinent to the plays written during her reign. Plays which contain these themes that are particularly striking include The Merchant of Venice, critically believed to have been written and performed in 1596-7, and A Midsummer Night's Dream, most likely performed in 1595. Elizabeth's rule is particularly pertinent in this respect as she never married despite considerable pressure to do so, not least early in her reign when this pressure came from the widower of her deceased sister, King Phillip II of Spain. Having been rejected, Phillip attempted to lead a large invading force on England via the Spanish Armada, a large invasion fleet. The armada was, however, defeated despite their far greater numbers, cementing Elizabeth's popularity amongst many and making English victory another popular theme in Shakespeare's plays, perhaps most markedly in 1599's Henry V which dramatises the victory of Henry V, another English monarch, against great odds at the Battle of Agincourt.
The other monarch under whom Shakespeare wrote his plays was James I of England. Before taking the English crown in 1603, James had been James VI of Scotland and was the first to unite the two thrones. Macbeth (1606), also known as 'the Scottish Play' is cited by Peter Ackroyd, a famous modern biographer of Shakespeare, as having its setting during a period of enthusiasm for Scotland that came after James's accession to the English throne. Given that Elizabeth I died unmarried and without a direct heir, some of the anxiety of the population about who would take the thrown upon her death might be imagined as a sub-text to another of Shakespeare's tragedies, King Lear (1605-6) which tells of the bloody civil war which struck ancient Britain after an ageing monarch stepped down from the throne.
During the life of William Shakespeare there were two monarchs that ruled England. The first was Henry the eights and the second was Elizabeth the first. Both were impressed with Shakespeare and commissioned him to write plays. This is one of the conditions of his success. There is alos some evidence that he was a member of a traveling theater group and did some teaching. In 1594, he became an actor and playwright for Lord Chamberlain’s Men. In 1599, he became a part owner of the prosperous Globe Theater.
Queen Elizabeth and King James
Elizabeth the First (reigned from 1558 to 1603)
James the First (reigned from 1603 to 1625)
Shakespeare lived from 1564 - 1616
(for the record, Henry VIII was 1509 - 1547)