What is the definition of point of tangency? in geometry

A tangent is basically a line that 'just touches' a curve - i.e. it is in a place where it approaches the curve, and a single point on the line touches a single point on the curve. It's quite an awkward definition since 'just touches' means only one infinitely small part of the line touches the curve, but it's quite obvious when you see it.
For that reason, I have attached a little diagram I improvised just now to help explain it. The red dot is where they meet, and the straight line is tangential to the circle.