Basic control theory, any junior/senior college class should suffice...
Here's a quick simplified hand-waving crack at it:
LTI = constant linear relationship between input and output independent of time -- the old y = mx + b bit where m (slope) and b (intercept) are constants and no matter when you measure the result is the same (same result if you do it now or some other time, just a shift in time). This can be applied to time-varying signals (like audio -- it is the system response that is time-invariant), and means if y =ax and y = bx then y = ax + bx (i.e. linear mathematical relationships like superposition and scaling can be applied).
Causal = output follows input based upon current and past values, no dependence upon future values, no output not related to the input.
Stable = just like it sounds, does not generate unbounded outputs (like oscillations) for any input.
Inverse may be a little trickier... If y = H1(x) where H is some function of x, then the inverse H2(y) = x; it is the function needed to generate x from y instead of y from x.
Mathematically the definition is unintuitive without the afore-mentioned class(es): a minimum-phase function sampled-time (z) domain has all its roots (poles) inside the unit circle defined by z = 1 (ensuring stability); a maximum-phase filter has all its zeros outside the unit circle. This for functions like y = [(x-z1)(x-z2)...]/[(x-p1)(x-p2)...] where zn are zeros and pn are poles.
Maximum-phase filters are also useful; the usual example is an all-pass filter, i.e. a filter that delays all frequencies equally (a pure time delay).
Off the cuff with a couple of lookups so somebody should check my memory...