The semicolon is fantastic; it can be used to join together two related sentences. See what happened there? I could have written, with perfect grammar, "The semicolon is fantastic. It can be used to join together two related sentences." But the subject in both sentences is the same. The second sentence flows on from the first. We can stitch these independent clauses together. (We could do something similar with the three sentences you just read. Notice how they are a little stilted?)
Imagine reading the sentences out loud. The semicolon takes a shorter breath than the full stop. The words flow with a better pace.
This usage is also appropriate to help prevent ambiguity. In a more complex paragraph the word "it" might potentially refer to more than one noun. The rewritten form makes it obvious that "it" refers to "semicolon".
The second use case for the semicolon involves the comma. Normally we use the comma to separate phrases that are not independent. But what happens when the phrases themselves include commas? Here we should use the semicolon to separate the phrases, avoiding ambiguity. Consider the next two sentences:
"Robin is a sound artist and composer who has recently delivered papers in Viseu, Portugal, Limerick, Ireland, and Kent, UK."
"Robin is a sound artist and composer who has recently delivered papers in Viseu, Portugal; Limerick, Ireland; and Kent, UK."
Knowing when to use the semicolon is largely a matter of style. There is rarely a case where it is essential, but it helps make your writing less tedious. And how do you develop your own intriguing writing style? Well, by reading a lot and writing a lot. You can start with the 650 articles here on this blog!