A complex sentence is a sentence consisting of an independent clause and a subordinating clause. Or, to put it more simply, complex sentences involve more than just a straightforward noun + verb. Complex sentences include extra phrases added to a base sentence, but which cannot stand alone as a sentence.
For example:
Simple sentence: I cannot work in the garden.
Complex sentence: Because it is snowing outside, I cannot work in the garden.
"Because it is snowing outside" is a dependent clause, and cannot stand on its own.
Another example:
Simple sentence: I intend to work in my garden this afternoon.
Complex sentence: If the weather stays fine, I intend to work in my garden this afternoon.
A complex sentence is one that contains a single independent clause and at least one dependent clause.