I play piano. Sometimes, measures in pieces are separated by a double vertical line instead of a single one. What does this signify?
In addition to indicating the end of a distinct section of the piece, a change in key signature, time signature or major tempo change, the double bar is also used to mark the location of a Da Capo or Dal Segno (a notation system that marks the repeating of a certain section of music without requiring additional measures to be written/printed.)
It is also used when a "fine" (end) of the piece appears in the middle of the piece. These examples are described and illustrated on the following website:
Adding to what Stephen Hazel said, a double-line can also indicate an abrupt change in key, key signature, tempo, or time signature or metrical phrasing or a different rhythm. It is basically a cue to alert the musician that one section of the piece is ending and a new section that is substantially different is beginning.