A semicolon provides a pause in a sentence that is longer than a comma but shorter than a full stop.
There are a number of ways in which one can use a semicolon to good effect; they are very versatile little punctuation marks.
A semicolon can be used to extend a sentence using two closely related ideas without using a conjunction. Therefore, a semicolon works differently than a colon does: they are not interchangeable. Examples:
- A semicolon is a much misunderstood piece of punctuation; while many people avoided them, those who use them achieve greater eloquence and refinement in their writing.
- You may have as many donuts as you like; however, the lemon meringue pie is all gone.
A semicolon can also be used to create a complex list in which each entry is accompanied by additional information before the list continues. Examples:
- When editing, I have to remember many things: my punctuation, because that tells people how to read my work; my spelling, because words are often confusing; my paragraphs, because they enable me to organise my ideas; and my word choices, because some words are really similar!
- The guests came from Melbourne, Australia; Paris, France; Montreal, Canada; and various other places.
How To Use A Semicolon
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WordyNerdBird 
Funnily enough I think I would have used a colon in your first example, ie “A semicolon is a much misunderstood piece of punctuation: …”. Not that a semicolon is wrong there, but the semicolon suggests to me that the two clauses (or phrases or whatever) on either side of it have the same ‘status’, whereas a colon can suggest that what follows it in some way explains or exemplifies or illustrates what preceded it.
I did consider the colon, but the semicolon works there nicely.