A woman who has a sexual relationship with a married man is called a mistress, right? But what do you call a man who has a sexual relationship with a married woman? I would like to know formal and informal ways to call such men in that regard.
"Mistress" (in regards to a sexual relationship) can mean either:
A "kept" woman, whose living expenses are paid by a man in return for her (usually exclusive) sexual favors. She may be paid money or given expensive gifts in addition to such expenses, in which case she may also be called a "courtesan" (although that rather suggests a non-exclusive relationship).
A woman of some social status who accepts a man as an unmarried sexual partner. There may be no payment involved, or she may even pay him. She is usually but not always married to someone else.
In both cases a long-term relationship is implied, and the term "mistress" is not usually used unless the man is of upper class, or at least above average social status.
In either case the man may be either married or unmarried, but in neither case are the two parties married to each other.
(If I am not mistaken sense 2 is actually older, darting back to the middle ages "courtly love" tradition.)
The term is somewhat out of date, I see few uses later than the 1970s.
There is no clear male equivalent for either sense that I know of.
"Gigolo" implies a male prostitute, and includes male equivalents of sense 1, but also includes shorter-term arrangements. It is also now somewhat obsolete, and was always offensive.
"Lover" can be used for anyone who has a sexual relationship, usually without marriage, with anyone else, but does not imply the specific social situations that "mistress" does.
"Adulterer" (or "adulteress") can be used for anyone who has a sexual relationship with a person who is married to another, but again does not always imply the specific social situational that "mistress" does.
There are many slang terms for prostitute and lover, carrying various implications, but none that carry quite the sense of "mistress" for a man, probably because when the term was devised, social and sexual situations between men and women was not at all symmetrical, and the word still carries those assumptions.
A mistress tends to be a woman that is "kept" by a man and implies a degree of inequality - ie the man has a double-life and the mistress is kept waiting. He may even be financially supporting her.
There isn't really an exclusively-male equivalent. "Lover" can apply to either sex in an affair, but equally two people in an honest relationship can also be called "lovers" without any negative connotation. "Paramour" can be used for either sex and is more exclusively used for illicit relationships, but is not widely used.