When is it appropriate to use a semi-colon?

Semi-colons should be used in order to join together two independent clauses together in order to show closely related ideas, for example: "My dad plays the ukulele and my brother plays the flute; I come from a very musical family." 

They can also be used to link together lengthier clauses which contain commas, or in a list if the items themselves include commas, in order to avoid confusion; for example, this sentence.