This is a good question. There are two basic points that need to be examined to answer your question. First, we need to define what a clause is. Second, we need to examine what a finite verb is.
Generally speaking a clause can be defined as something having a verb. An independent clause is a complete sentence and the dependent clause is subordinate to the main sentence.
"Jim walked" is Independent clause.
"While Jim walked, he saw a bird" is a dependent clause, as the clause "While Jim walked" depends on the main sentence.
A finite clause is a clause that has a verb with tense. The opposite of this would be a clause that has an infinitive (non-finite clause). The key is the tense of the verb. Let me give a few examples.
These two sentences are examples of a finite clause: "Jim played the piano" and "As Jim's mother called him, he played the piano."
An non-finite clause is as follows: "Jim loves to play the piano." Notice the verb "to play" is in the infinite, which does not have a tense.