Parliament wanted Elizabeth to marry for two reasons:

To produce an heir to succeed her to create stability. This was especially the case after Elizabeth caught smallpox in 1562 and nearly died.
To build a powerful alliance with another country.

Elizabeth became angry with Parliament pressuring her and so stopped them from discussing her marriage after 1566

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Elizabeth's main potential husbands were:

Robert Dudley, Earl of Leicester

Dudley was Elizabeth's favourite, they were possibly lovers. However, it was impossible for Elizabeth to marry him after his wife died in suspicious circumstances.

Francis, Duke of Anjou & Alençon

Brother of the French King. Marriage to Francis could have created a powerful alliance, but he was Catholic so he was unpopular in England. There were also worries that, should Elizabeth die childless, the French monarchy might takeover England.

Philip II of Spain

Marriage to Philip would have created the most powerful alliance in Europe. However he was also Catholic and Mary I's marriage to him had sparked Wyatt's rebellion in England. However Elizabeth kept Philip believing that he was a possible match to help diplomatic relations with Spain. Elizabeth's refusal to marry Philip, though, was one of the causes of his failed Armada of 1588.

Elizabeth never found an ideal husband. She also didn't marry because she didn't want a husband to control her. In addition, she worried that if she married one English nobleman it would upset the others. She might also have disliked marriage because of the disastrous example of her parents' marriage, with her father executing her mother.

GCSE Elizabethan England 1568-1603 AQA

The Marriage Question