Is the underlined clause independent or subordinate? watch the babies while mother takes a nap.

The sentence has two clauses: "Please watch the babies" and "while Mother takes a nap". (You didn't underline a clause so I'm going to explain both of them).

First Clause: "Please watch the babies"
This is an independent clause. You can identify independent clauses when you read it and it has a full idea already. It can stand without another clause to back it up.

Second Clause: "while Mother takes a nap"
This is a subordinate clause. Subordinate clauses usually start with a conjunction. You'll see that the given clause starts with "while", which is a conjunction.