Clauses are meaningful groups of words built around a verb.
The minimum requirement for a clause is that it should have a verb (an action) and someone or something to implement that verb (carry out the action). In most cases, it will also have the thing on which the action is carried out. Functionally, this is subject + verb+ object or one of its variants like subject+ verb+ complement. There are more complex ways of dividing up the language structure, but SVO or SVC give us all that we need for this topic.
A sentence is a clause on its own, but other clauses can be tacked on if they deal with a common topic.
There are several classes of clauses.
(1) A main clause is capable of standing on its own two feet and forging an independent and high-flying career as a sentence: test it by inserting a full stop and see if it looks like a sentence.
(2) When two main clauses are joined together to form a sentence, they tended to need careful handling to stop them fighting. Usually one will be given the job of being the main clause and the other will be linked to it because they both share the same topic. If both clauses are equally important to each other they will usually be classed as coordinating clauses, which settles the seniority problem by giving them the same rank. In formal punctuation the link between such strong clauses will be made by using a colon, although in more modern usage the default is to use a comma.
(3) A subordinate clause needs to borrow the subject of its main clause to make sense in the sentence. It hasn’t got enough information to be a sentence on its own and will always be dependent on mummy or daddy. Bummer. Subordinate clauses need the reader to mentally connect up the sense of the main clause/s before making sense of the whole sentence. In formal punctuation the link between main and subordinate clauses will sometimes be made by a colon, more commonly by a semi-colon; and in modern usage the default is to use a comma. A conjunction can often do the same connection job just as well: it’s a matter of style, not grammar.
Phrases are groups of words which have an understandable meaning, but miss out on being sentences because they have no verb.
Phrases only get connected to a verb if they are added to a sentence (which by definition has a verb) and are then governed by the action of that verb. Commas often make the connection, but commas can also separate phrases out from the main clause/s for stylistic reasons. Phrases have no life of their own: if they don’t hang around with their verbs and behave themselves they become the street kids of Grammopolis, loose and lonely with no home to go to.