A prince consort, generally speaking, is the husband of a Queen regnant, unless he himself is a king.
Current examples include the Duke of Edinburgh (husband of Queen Elizabeth II of the United Kingdom), and Prince Henrik of Denmark (husband of Queen Margrethe II of Denmark).
In recognition of his status, the prince consort can be given a formal title, such as Prince, Prince Consort (see below) or King Consort, with Prince being the most common. However, most monarchies do not have formal rules on the styling of princes consort, so it is quite feasible that such a person may be left entirely without royal titles. It is also not fully clear what would happen to the prince consort's title in the event that he were to outlive his wife, as this seems to have rarely occurred in the recorded annals of monarchy. In a recent example, the title of Prince Bernhard of the Netherlands did not change after the death of his wife, Queen Juliana (reigned 1948-1980, Queen Mother 1980-2004).
Prince Consort (capitalized) is quite different from the above as it is a formal title. Prince Albert of Saxe-Coburg-Gotha is the only British prince consort to have ever held it. It was awarded to him in 1857 by his wife, Queen Victoria (reigned 1837-1901), in recognition of his close to co-regency status.
The female counterpart of the title, Princess Consort, has also never been used, as wives of British Kings have always been Queens. However, it has been said that when the present Prince of Wales becomes the sovereign, his wife will not be known as The Queen but rather as The Princess Consort (see Duchess of Cornwall and Princess Consort).