A colon (:) is frequently used to introduce a list or an example. So, you could say something like "These are the fruits that I can think of off the top of my head: apple, banana, pear, orange, guava, watermelon."
A semicolon divides two independent clauses... another way to think of an independent clause is a complete thought... something that could stand on its own. You use the semicolon to combine those two thoughts/sentences/clauses to show that there is a closer connection. "Bob was sad. His dog died." becomes "Bob was sad; his dog died."