Definition of the Orthocenter of a Triangle
If the triangle is obtuse, such as the one on pictured below on the left, then the orthocenter will be exterior to the triangle.
If the triangle is acute, then the orthocenter is located in the triangle's interior
What about the orthocenter of a right triangle?
The illustration above demonstrates that the orthocenter of an obtuse triangle is situated in the triangle's exterior; while an acute triangle's orthocenter is located in the interior. So the question is, where is the orthocenter located in a right triangle?
The orthocenter is actually concurrent with the right angle! In other words, the orthocenter is located where the right angle's vertex is (see red point in the pic below)