For questions specifically about existence (feasibility), counting feasible points (in a discrete setting), or partitioning the feasible region (e.g. in the context of branch-and-bound methods). Feasible points are those which satisfy all the constraints of a problem, and so can be defined without regard to any optimization criterion.
In constrained optimization problems the feasible points are those which satisfy all the constraints, and thus are candidate solutions to a problem before evaluation of an objective function narrows down the possibilities.
If the constraints are linear and the variables are continuous, then the feasible points will form a (not necessarily bounded) convex region. If in addition the objective function is linear and the feasible region is bounded, then the maximum and the minimum values of the objective function will be attained at a vertex of the feasible region. This is the geometric foundation of the simplex method for solving a linear program. In practice the simplex method usually locates such feasible points efficiently, but rigorously proving the complexity of the linear feasibility problem is polynomial led to study of the Ellipsoid Method and the Interior Point Methods.
If the constraints are linear and the variables are discrete (integer-valued), then deciding feasibility is generically difficult, and is NP-complete even when variables are restricted to binary 0-1 choices.