A clause is a combination of related words containing a subject and a verb. It can be a simple sentence or a part of a compound sentence.   

Examples:

Clauses have two major types:

1. Independent Clause (Main Clause)
2. Dependant Clause (Subordinate Clause)

Read the following sentence:

I saw a man who was crying.

The above sentence has two clauses: ‘I saw a man’ and ‘who was crying’. The first clause ‘I saw a man’ gives a complete meaning and can alone stand as a complete sentence. Such a phrase is called Main or Independent clause.

On the other hand, the second clause ‘who was crying’ does not give a complete meaning and cannot (as alone) stand as a complete sentence. It depends on the main clause to give a complete meaning. Such a clause is called Subordinate or Dependent clause.

Main clause or Independent clause is that clause which expresses a complete meaning. It alone can stand as a sentence.

Examples
I saw a man who was crying.
The professor asked many questions but no one could answer.
I met a friend who helped me a lot.
They contacted the customer who had not paid the bill.
He does not like the people who smoke.
We met a man who could speak many languages.
She loves her husband, who never tells a lie.

Subordinate clause or Dependence clause is that clause which (as alone part) cannot express a complete meaning. It alone cannot stand as a sentence because it depends on the other clause (independent clause) to give a complete meaning. It serves a subordinate role in the sentence. 

Examples:

I saw a man who was crying.
The professor asked many questions but no one could answer.
I met a friend who had helped me a lot.
They contacted the customer who had not paid the bill.
He does not like the people who smoke.
We met a man who could speak many languages.
She loves her husband, who never tells a lie.

Dependent Clauses have further three types:

1. Noun Clause
2. Adjective Clause
3. Adverb Clause

To read the types of subordinate clauses, click on "Types of Subordinate Clause"