Semi-colons can be used to join two separate sentences that are on the same theme.
It was bitterly cold. Josie's nose was tingling.
These sentences are fine as they are, but because the meaning of the second sentence is closely linked to the meaning of the first one, we can join them with a semi-colon.
It was bitterly cold; Josie's nose was tingling.
It is not correct to join two sentences like this with a comma. This is a very common mistake.
It was bitterly cold, Josie's nose was tingling. (wrong)
A semi-colon can replace a conjunction in compound and complex sentences.
My mum is forty but my dad is forty two.
My mum is forty; my dad is forty two.
Both these sentences are correct, but using a semi-colon adds variety to writing, making it more interesting. The reader understands the word 'but' even though it is not actually there.
Again, it is not correct to use a comma to join two clauses in this way.
My mum is forty, my dad is forty two. (wrong)