From The Grammarly Blog:
What is a semicolon?
Semicolons (;) are as basic as a period stacked on top of a comma. Does that mean you can use it like either one? Don’t get your hopes up. But don’t let this punctuation mark get you down either.
How to use a semicolon correctly
The most common use of the semicolon is to join two independent clauses without using a coordinating conjunction like and .
Do you use a capital letter after a semicolon? The general answer is no. A semicolon should be followed by a capitalized word only if the word is a proper noun or an acronym.
We can go to the museum to do some research ; Mondays are pretty quiet there.
Remember, semicolons are not interchangeable with commas or periods. Instead, they’re somewhere in between: stronger than a comma but not quite as divisive as a period. Sounds pretty cunning to us.
Here are the rules for using semicolons correctly; we hope you’re taking notes.
1 Use semicolons to connect related independent clauses
You can use a semicolon to join two closely related independent clauses. Let’s put that another way. The group of words that comes before the semicolon should form a complete sentence, the group of words that comes after the semicolon should form a complete sentence, and the two sentences should share a close, logical connection:
I ordered a cheeseburger for lunch; life’s too short for counting calories.
Martha has gone to the library; her sister has gone to play soccer.
The examples above are each made up of two complete, grammatically correct sentences glued together.
That’s exactly why you can’t substitute a comma for a semicolon. Using a comma instead of a semicolon in the sentences above would result in a comma splice. And there’s nothing as painful as a comma splice.
2 Skip the coordinating conjunction when you use a semicolon between two independent clauses
A semicolon isn’t the only thing that can link two independent clauses. Coordinating conjunctions (that’s your and s, but s, and or s) can do that too. But you shouldn’t use a semicolon and a conjunction. That means that when you use a semicolon, you use it instead of the and s, but s, and or s; you don’t need both.
Here’s a hint: You know how you can use a comma and an and to link two related ideas? Think of the period that forms the top part of the semicolon as a replacement for and .
I saw a magnificent albatross, and it was eating a mouse.
I saw a magnificent albatross; it was eating a mouse.
You need a comma plus something to avoid a comma splice. That something can either be the right conjunction or the period that turns a comma into a semicolon.
A semicolon can replace a period or a comma and a coordinating conjunction to demonstrate contrast between independent clauses instead of agreement. This is part of the same rule, but the conjunction in question is but instead of and . In other words:
This is part of the same rule; the conjunction in question is but instead of and .
To summarize, a semicolon links up two related ideas by narrowing the gap between the ideas of two separate sentences or by replacing a coordinating conjunction between the ideas. That goes for showing contrast too: just because two ideas are opposed or contradictory, that doesn’t mean they aren’t related closely enough to earn themselves a semicolon.
Link to the rest at The Grammarly Blog

