A line that touches a circle at only one point. Also, when someone goes off topic and starts talking about something else.
The math meaning is about shapes and lines. The other meaning is about talking and not staying on the main idea.
Examples:
The tangent touches the circle at only one point.
She went off on a tangent about her cat during the meeting.
The tangent is a line that touches the circle.
The tangent of the circle is not very long.
During the lesson, the teacher went off on a tangent about her vacation.
He drew a tangent to the circle and then talked about his dog.