It's much simpler than the apostrophe, really it is. But it needs to be considered in the context of other possibilities and that may seem complicated. It's not.
Think for a moment of a ball rolling down a ramp, and then coming to a stop at a barrier. If the sentence is that ball, it runs straight ahead down the ramp and stops when it bumps into a full stop: a period.
"I got up this morning."
You can write (rather jerkily) using nothing but short sentences, short ramps one after another: "I got up this morning. That was my first mistake."
If the ball, instead of rolling smoothly down a straight ramp, instead rolls down stairs with a landing in the middle, where it changes direction, it will bounce off the walls before going on down and there will be some bumps on the way. The two half-flights of stairs are complete--they have a top, bottom, and all the steps in between. As a sentence this model defines the combination of two complete thoughts squeezed into one sentence--where each part could be a short sentence. The bumps on the way are where you need commas. The bouncing on the wall of the landing is where you need either a conjunction, a semi-colon, or (in extreme cases) a colon.
"I got up this morning and that was my first mistake." The same two complete thoughts, grammatically complete sentences, joined by a conjunction. It feels easy to read, easy to understand; it's uncompressed, relaxed, open. When you want to slow down or relax a passage, the conjunction is your friend. But too many of them in a row (like too many short straight "ramp" sentences in a row" feel amateurish. Once small children learn they can string thoughts together, they talk like that "I went to the zoo and I saw a tiger and I had an ice cream and I saw ducks and geese and I had a hot dog and..." What we have here is not really the staircase with its landing, but a curving slide that can (if you're not careful) go on forever.
"I got up this morning; that was my first mistake." Here's the semi-colon. It signals the reader to expect a complete clause--subject, predicate--on each side of the semicolon, but it tells the reader (whether reading aloud or silently) not to stop...to leap over the semi-colon and connect the meaning of the second part to the first. It's the equivalent, in driving, to a rolling stop. Thus it is faster to read, it feels more compressed, more logical, even (depending on the rest of the sentences so joined) a bit abrupt.
With care in crafting the sentences in front of it and behind it, a semi-colon can also make the whole more formal, less formal, more compressed, or more rhythmic. It gives the writer flexibility in sentence length, tone, compression/extension, pacing...all good things to have. Like any other tool in the writer's toolbox, it can be overused--but it's essential.