A clause is a part of a sentence. For example, "The sun shone." is a sentence with one clause, and Samuel Beckett's famous "The sun shone, having no alternative." is a two clause sentence. The Beckett example also shows the grammatical subordination of one clause to another - "having no alternative" is grammatically incomplete, being dependent on the main clause "the sun shone" for its meaning, and so is understood as less important than this main clause. If the example was modified to read "The sun shone, and Murphy broke into a sweat." then there would no longer be a subordinate clause, but two parallel clauses, since neither clause depends on the other for its grammatical completeness - the example could easily be broken up into two separate sentences, "The sun shone." and "Murphy broke into a sweat."
When editing your writing, for each sentence you need to consider both the number of clauses, and whether they are subordinate or parallel. More than two parallel clauses in a sentence will normally create problems for your reader, since they will allow for a greater number of possible inferences than can easily be carried in the reader's head. Subordinating one clause to another through grammar indicates a hierarchy of importance akin to outlining, and so provides your reader with a much greater sense of coherence. But even with a careful use of subordinate clauses, it can be considered one of the rules of modern academic writing not to put more than three or four clauses in a single sentence.