Generically means something that satisfies conditions. One example is the admissibility of a direction of change of some parameter, such as a right-hand side of a linear program (LP) from $LaTeX: b$ to $LaTeX: b + t h$, where $LaTeX: t$ is some scalar and $LaTeX: h$ is the direction of change. For a positive value of $LaTeX: t$, this might cause the LP to become infeasible, no matter how small it is. In that case it is common to call the direction vector inadmissible. For a perturbation to be admissible in this context, it is required that the mathematical program have a solution in some neighborhood of that direction. For the example, $LaTeX: h$ is admissible if the (primal) LP is feasible for all $LaTeX: t \in [0, t^*)$ for some $LaTeX: t^* > 0$.
Admissibility need not be for directions of change in the context of sensitivity analysis. The term is used elsewhere and defined for the context in which it is used. For example, a primal ascent algorithm could define a direction, say $LaTeX: d$, to be admissible if $LaTeX: x + td$ is feasible and $LaTeX: f(x+td) \ge f(x)$ for all $LaTeX: t \in [0, t^*)$ for some $LaTeX: t^* > 0$.