A clause is defined as a group of words with a subject (noun phrase) and verb phrase.
An independent or main clause contains a subject (i.e., noun phrase) and verb phrase and can stand alone; that is, it states a complete idea: The baseball player is running around the bases.
A compound sentence contains two independent clauses that is separated by a coordinating conjunction: The baseball player is running around the bases, and her helmet falls off rounding third.
A complex sentence contains one independent clause and one dependent clause (refer to Dependent Clauses): The baseball player is running around the bases while the crowd cheers with excitement.
In a compound sentence and complex sentence, the terms clauses and sentences are not the same. For example, in a compound sentence, there are two independent clauses in a single sentence; in the complex sentence the independent clause is The baseball player is running around the bases but the sentence refers to both the independent and dependent clauses (i.e., The baseball player is running around the bases while the crowd cheers with excitement.).
The only time a clause and sentence are the same is when writing a simple sentence. A simple sentence has one (independent) clause: The baseball player is running around the bases.
A compound-complex sentence contains two independent clauses and one dependent clause: The baseball player is running around the bases, and her helmet falls off rounding third while the crowd cheers with excitement.
Subordinating clauses | Relative clauses | Nominal (noun) clauses