What is an inscribed angle, and how do I solve them in geometry?

In addition to Steveschoen's answer:
(1) The measure of an inscribed angle is 1/2 the measure of its intercepted arc. (In the diagram, 1/2 the measure of arc AC.)
(2) Inscribed angles intercepting the same or congruent arcs are congruent. (In the diagram, choose any point X not on arc AC; the measure of angle AXC is the same as the measure of angle B.)
(3) Tangent-chord angles are closely related as their measure is also 1/2 the intercepted arc.

Inscribed angles are angles inside circles where the vertex is "on the circle", as in the attachment, the angle in red. The green angle is called a central angle.As for solving them, that would depend upon the problem. I can tell you that the measure of the inscribed angle is half the measure of the central angle. So, for instance, if the central angle was 60 degrees, the inscribed angle would be 30 degrees (1/2 * 60). Then, the opposite way, if the inscribed angle was 25 degrees, the central angle would be 50 degrees (2 * 25).