Just a few words on what you can expect when you follow T. Breuer's advice and study a textbook on signals and systems: 

Consider a 'system',  with one input continuous time signal and one output continuous time signal.  Suppose the system is linear (if you add two input signals or multiply them by a number, the output will be again a sum or a multiple) and time invariant (it behaves the same at all times) -- "LTI-S". This happens typically for systems of springs and masses (mechanics) or coils, condensors, resistors (electronics). The mathematical model for such systems is differential equations. The good news is that everything becomes easier by a certain 'trick'.   There is a transformation of the signals  (by means of an integral), so that each signal is represented by a function of one complex variable. In many cases this transformation (in both directions) can be performed without actually calculating integrals, but by means of a 'translation table' and a few rules.

Now, what does the LTI-S do to the transformed signal?  Nicely the system multiplies the transformed signal with a certain function of one complex variable -- the transfer function.   In the easy cases this is a rational function.  Even though you will have to learn the relation between the position of these poles and stability from a text-book, the essence of this relation is easily explained:   When an  LTI-S is described by an ordinary differential equation with constant coefficients, solutions can be obtained -- in a sense - by a superposition of fundamental solutions.  A pole p corresponds to a fundamental solution of the form e^(pt). If the real part of p is positive this solution grows beyond all bounds with increasing t, if it is negative it will become arbitrarily small.  The exact mathematics has to be done, but it may be plausible that a fundamental solution which grows exponentially can produce arbitrarily large outputs from bounded inputs.  This would correspond to one definition of instability.

So, you would need mathematics, but thanks to the transformation, basic knowledge about functions and little or  no calculus to just *use* the theory.  For understanding you would need calculus.

In simple I can tell you that poles will bring transfer function to infinity while it is set to 0. On the other hand, zeros will make transfer function to be zero while it is set to 0.