Fornication is pejorative term for sex between two people who are not married to each other, as it implies the sinful pursuit of physical pleasure.
Adultery has various definitions depending on local convention, but most often refers to sex between a married person and a person who is not his or her spouse, normally without the consent (or knowledge) of the spouse.
The terms overlap. Someone can be both a fornicator and an adulterer, as the first may be considered immoral, while the second can be considered both immoral and illegal (or, at least, may have legal consequences).
In some cultures, religions, and regions, the definition of adultery covers any sexual intercourse between unmarried persons, and in this case any separate accusation of fornication is redundant.
Final note: You would not accuse someone of fornication unless you strongly believe that premarital (or extramarital) sex is sinful, as the term has religious overtones. If instead you feel someone engages in gratuitous sex, or is overly careless with their affections, there are various other expressions which fit better.