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.