Persistence is the ability to stick with something. If you practice the violin for over a year in order to play "Twinkle, Twinkle, Little Star" perfectly, that's persistence!
Persistence can also mean something that lasts for a very long time. The persistence of bullying, despite years of efforts by the school, has frustrated many parents. Bed bugs are known for their persistence. Despite near eradication in the 1950s, they are back and harder to eliminate than ever.