Once upon a time, back when writing was new, commas and such were used mainly as cues to tell people when to pause for breath. (Ever see those medieval libraries with cubicles everywhere? I imagine that's because everyone was muttering aloud as they read! Silent reading developed with education.) However, as writing began to be used for more than just reading aloud, punctuation began to fulfill grammatical functions as well. The most important of these is delineating clauses, and this is where the semicolon comes in.
(If you know your clauses, you can skip to the bottom.) There are two types of clauses: Independent Clauses (IC), and Dependent Clauses (DC).
An IC has a subject and a predicate (verb + optional object), and can "stand alone". "Stand alone" means that if you walked into a room and said it, people wouldn't look at you like you were too crazy. (I suppose that depends on the content as well, but ignore content for a second!)
"I went to the store." = IC
The subject is "I", and the verb is "went".
"Went to the store" is NOT AN IC! There's no subject.
A DC depends on the IC for it's meaning.
"She bewailed her fate, crying on his shoulder."
"She bewailed her fate" = IC
"crying on his shoulder" = DC
If you just walked into a room and said "Crying on his shoulder," people would be very confused. It's dependent on the rest of the sentence for meaning.
End Preamble... On to semicolons! A semicolon comes between two IC's in one sentence.
"She bewailed her fate." + "Life was so cruel." =
"She bewailed her fate; life was so cruel."
"I've always loved this company; I hope I can work here."
"Thank you for your hospitality; I really appreaciate it."
There is another use for semicolons, but it's quite rare. If you have a list of items, and each item has a comma or two in it, you can use semicolons to avoid confusion:
"I asked for three things: the ability to choose my own hours, regardless of the convenience of the company; the right to dictate my own benefits plan, including life insurance and dental; and the right to dress casual, even at company-wide meetings. They decided not to hire me."
Hope this helps!