"I enjoyed the apple pie that you bought for me." Here, "I enjoyed the apple pie" is an independent clause and "that you bought for me" is a relative clause. The independent clause could stand alone as a simple sentence without the relative clause.
"I ate breakfast before I went to work." This has "I ate breakfast" as an independent clause, and "before I went to work" as a dependent clause.
Examples of sentences that have more than one clause but are not complex sentences include the following:
"I was scared, but I didn't run away." Both of these clauses are independent in this compound sentence.
"The dog that you gave me barked at me, and it bit my hand." Here a compound-complex sentence has two independent clauses ("The dog barked at me" and "It bit my hand") and one dependent clause ["that you gave me"].