A clause is usually defined as a subject + verb, which, to me, is a syntactical definition. It's like: OK, so that's how it looks, but what does it mean? And, what's the relationship between a clause and a sentence?
A sentence consists of one or more clauses and it makes a statement (or expresses a desire or emotion, asks a question, or gives a command). A clause can do those things too, but it does not have to; a clause can simply supply supplemental info to one of them. A clause contains a verb, explicitly or implicitly.
Entering the room ...
Entering the room, I was surprised to see a cat on the mat.
taking this here bucket
Run down to the creek, taking this here bucket, and get me some water to make us some moonshine. Okay?