Fleur's Mum had had a fling with someone in the past - hence Fleur who ended up spurned and left a curse mark on Apolline, the mum.
Apolline met Mr Delacour, who is possibly a muggle, and eventually convinced her to have a second child, Gabrielle. The curse mark flared up and Apolline died in childbirth. Mr. Delacour ended up depressed so Fleur basically raised Gabrielle.
She developed an interest in curse marks, especially in Harry's, as that one was idle longest, and an ambition to become a curse breaker herself, for which she needs extra money, which is where the Triwizard comes in