Joining Two Clauses
The most common explanation given for semi-colons is that they are used to join two different clauses (sentences) that could exist perfectly understandably separately.
Example: I bought a new coat yesterday. It's so soft. I bought a new coat yesterday; it's so soft.
Both sentences make perfect sense alone, but joining them puts more emphasis on 'it', making it all the more clearer that 'it' refers to the 'hat'.
However, this definition is rather foggy… Surely, a lot of sentences can be treated like this?
A much more precise definition is that semi-colons always replace conjunctions (connectives). These are, inter alia: because, and, however, or, yet, so, nor, instead…
Most commonly, the semi-colon is used to replace the conjunctions 'because' and 'and'.
Examples: I hate this new bed because it's so uncomfortable. I hate this new bed; it's so uncomfortable. Call us tomorrow, and we'll tell you then. Call us tomorrow; we'll tell you then.
Semi-colons can also introduce conjunctions, rather than replace them, if an emphasis of the conjunction is desired, or if the conjunction plays a vital part in our reading of meaning.
Examples: There are many activities to do at the resort; for example, I did skiing yesterday. Sharon is a really nice person; however, I don't particularly like her husband. He was a very talented artist; and he could do plumbing, too!
Emphasis in Juxtaposition
Semi-colons can also join two clauses that directly oppose each other (we call this an antithesis).
Example: The men went left; the women went right. The sky is blue; the grass is green.
Note again in these examples, the semi-colon can always be seen as replacing a conjunction.
You may have also noticed that semi-colons are used frequently in lists. However, this does not apply to all lists; semi-colons are used in complex lists only.
Here is the difference:
Simple List: I went to the shops and bought some oranges, a kiwi and some milk. Complex List: I went to the shops, and I bought some oranges, as I had run out; a kiwi, because Simon wanted one; and some milk. Simple List: For this DIY, you will need hand tools, power tools and correct attire. Complex List: For this DIY, you will need hand tools: a flathead and a crosshead screwdriver, a wrench and a saw; power tools: a cordless drill, a nailer and a circular saw; and correct attire: a hardhat, safety glasses, etc.
You'll notice that in complex lists, the listed items come with extra details or clarifications and that the semi-colons come after these.
NB: The final item in a complex list is preceded, too, by a semi-colon. In vertical lists, such as bullet-pointed lists, semi-colons are unnecessary and optional.