Since you didn't specify where, I will use England.
In the early 1500s, drama was evolving from the religious dramas of Middle Ages. In England there were the cycle plays which told the stories in the Bible. There were also the Miracle, Mystery, and Morality plays that grew out of these. The most famous was Everyman which appears at the end of the 15th century.
In the middle of the 1500s, three plays were written and became popular. First came two comedies, Gammar Gurton's Needle by Mr. S. (1533) and Ralph Roister Doister (1553) by Nickolas Udall. The first tragedy which was based on Seneca's closet dramas, was Gorboduc by two school friends, Thomas Sackville and Thomas Norton.
The next important development was the building of the Theatre by James Brubage. For the first time his acting company, The Earl of Leiscester's Men (later The Lord Chamberlain's Men and finally The King's Men) had a permanent home. Other theatres followed, IE the Curtain and the Rose.
With the establishment of the theatres, there was a need for new plays. By the end of the 1500s, a group of university educated men, called the University Wits began to write for the public theatres. Perhaps the two most important of these men are Thomas Kyd who wrote The Spanish Tragedy which became one of the most popular plays of the Elizabethan period and Christoper Marlowe. He proved to be the most skilled at using blank verse which became the standard verse form for plays during this period.
He was quickly followed by a host of other playwrights who continued to explore story telling using blank verse. The giant of this period today is William Shakespeare but there were many other playwrights including Ben Jonson, Francis Beaumont, John Fletcher, Thomas Middleton and Thomas Decker.
With the death of Elizabeth and the ascension of James I to the throne, the period is called the Jacobean. Theatres, both public and private continued to flourish until the Commonwealth, under Oliver Cromwell, when the theatres were closed until the Restoration. The Puritans who were in control believed that theatre were evil.