A structure is statically indeterminate if there are more unknown forces than the number of static equilibrium equations available to solve for those forces. For example, in 2D analysis of a beam, there are 3 static equilibrium equations: sum of F_x = 0, sum of F_y = 0, and sum of M_o = 0. Thus a simply supported beam with a pin at one end and a roller suport at the other, is statically determinate, because there are 3 unknown reactions (2 forces at the pin and one at the roller), and the 3 static equilibrium equations available to solve for those forces. A cantilever beam fixed at one end is also statically determinate (2 forces and a couple unknown at the fixed end, and the 3 static equilibrium equations available to solve for them). A beam on 3 simple supports, or a propped cantilever fixed at one end and simply supported at the other, would be statically indeterminate because you have more than 3 unknown forces, and you would have to resort to compatability/deformation equations to solve for the additional forces. A statically indeterminate truss is a bit different, in that while the suport reactions may sometimes be solved using the 3 static equilibrium equations, the member forces themselves framing into a joint may not be able to be determined using the standard equilibrium equations, for example, when you have multiple members framing into a single joint. There's a formula floating around somewhere that identifies to what degree a truss may be indeterminate, based on the number of members and joints, etc. , but they are too confusing for me to figure out.
