Semicolons are used in a variety of ways, but two common uses are to split up two complete thoughts within the same sentence and to split up groups of elements within a longer list.
Take for instance the following two statements: "He was very tired. He went to bed." The two thoughts are distinct, yet very closely related. While most often, someone would write "He was very tired, so he went to bed" you could also skip the conjunction and write "He was very tired; he went to bed" in order to give the going to bed part equal status with his being tired.
Semicolons are also used to break up groups of elements in a longer series. Basically, if an element of a series has its own commas, then semicolons replace the commas normally used between things. For instance, compare "He is survived by a son, John Smith, of Dallas, Texas, two sisters, Jane Doe, of Honolulu, Hawaii, and Susan Smith, of Richmond, Virginia, and a grandchild, Kenny Kant, of Cheyenne, Wyoming."
That's a lot of commas and confusion, making the sentence hard to read. Semicolons act as "super commas" here, providing thought breaks between groups of information. This reads more clearly: "He is survived by a son, John Smith, of Dallas, Texas; two sisters, Jane Doe, of Honolulu, Hawaii, and Susan Smith, of Richmond, Virginia; and a grandchild, Kenny Kant, of Cheyenne, Wyoming."