Q: Why did the dentist and the manicurist decide not to be friends?
A: Because they were always fighting tooth and nail!
Explanation: A dentist is a doctor that improves the health of your teeth. A manicurist is a specialist who improves the health and appearance of your fingernails.
To “fight tooth and nail” is an idiom that means to fight hard with all of your resources to achieve a goal. You might fight tooth and nail against an injustice. A union might fight tooth and nail for its members. Firefighters fight tooth and nail to put out a fire.
This joke is funny because it plays with the idiom, and the work that these two professions do: dentists work with teeth and manicurists work with nails.
Here is Foreigner singing about fighting tooth and nail. Some of the lyrics are hard to understand so here they are.