Simple

My friend invited me to a party. I do not want to go.

Compound

My friend invited me to a party, but I do not want to go.

Complex

Although my friend invited me to a party, I do not want to go.

In the first example, there are two separate simple sentences: "My friend invited me to a party" and "I do not want to go." The second example joins them together into a single sentence with the co-ordinating conjunction "but," but both parts could still stand as independent sentences -- they are entirely equal, and the reader cannot tell which is most important. In the third example, however, the sentence has changed quite a bit: the first clause, "Although my friend invited me to a party," has become incomplete, or a dependent clause.

A complex sentence is very different from a simple sentence or a compound sentence because it makes clear which ideas are most important. When you write

My friend invited me to a party. I do not want to go.

or even

My friend invited me to a party, but I do not want to go.

The reader will have trouble knowing which piece of information is most important to you. When you write the subordinating conjunction "although" at the beginning of the first clause, however, you make it clear that the fact that your friend invited you is less important than, or subordinate, to the fact that you do not want to go.