What's the word for the person who the cheater is with?

Solution 1:

Consider paramour:

a lover, especially the illicit partner of a married person.

Oxford Living Dictionaries

This word has connotations of illicitness that apply beyond the contexts of marriage. Someone in a serious relationship (but not married) might also have a paramour.

If it is a man cheating on his spouse with another woman, the go-to term is mistress:

A woman (other than the man's wife) having a sexual relationship with a married man:

Oxford Living Dictionaries

Here are some less satisfying alternatives:

First, homewrecker, which means:

[informal] A person blamed for the break-up of a marriage or long-term relationship, especially as a result of having an affair with one of the partners

Oxford Living Dictionaries

This term is slightly pejorative and skewed to the perspective of the person being cheated on, so it may not work in your context.

Second, side chick (or its variant, side piece):

The girl a guy cheats on his girlfriend with

Urban Dictionary

Both of these are rather informal, so I suspect they may not be what you're looking for.

Lastly, this person is sometimes (euphemistically) referred to as the other woman/man, with emphasis on other.

Solution 2:

The other woman/man. Although the "other woman" is more common than the "other man".

From M-W

(noun) a woman with whom a married man has an affair —usually used with the

"tired of being the other woman, she gave him an ultimatum: divorce his wife or take a hike"

Solution 3:

How about lover?

OD:

lover: A partner in a sexual or romantic relationship outside marriage: ‘I think she had a secret lover’, ‘they had been lovers for years’

Your example:

The cheater cheats on their significant other with their lover.

In context -- that is, given that one of the partners in a committed relationship is cheating on the other -- lover makes perfect sense: the cheater has, or has taken, a lover. Lover also has the advantages of being one word and gender neutral.