Trigonometry based on a unit circle and radians and trigonometry based on a right triangle.
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The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.
Angular displacements measured in radians or stradians, lengths of lines measured in units of length.
plane trigonometry spherical trigonometry
The main kinds are plane trigonometry and solid trigonometry. The latter will include trigonometry in hyper-spaces.
There are more than two but the two most common ones are plane trig and spherical trig.