Definition: The incenter of a triangle is the point where its three angle bisectors meet; it is equidistant from the three sides and is the center of the inscribed circle.
What is special about it?
The incenter is always inside the triangle. Its distance to each side is the same, and that distance is the radius of the incircle, the largest circle that fits inside and touches all three sides.
How do you find the radius?
The area of the triangle equals the inradius times half the perimeter, so r = Area / s, where s is the semiperimeter.
Example. A right triangle has sides 9, 12, and 15. Its area is (1/2)(9)(12) = 54. The semiperimeter is (9 + 12 + 15)/2 = 18. So the inradius is 54/18 = 3. The incircle has radius 3 and touches all three sides.
What do students mix up?
The incenter and the circumcenter. The incenter uses angle bisectors and sits inside; the circumcenter uses perpendicular bisectors and may sit outside. Remember that the in in incenter means the circle lies inside the triangle. Another error is to use the full perimeter instead of the semiperimeter in the radius formula.