Stability of Transfer Function H(s)
What are poles?

Poles are essentially the exact same as asymptotes. You've dealt with them a lot (hopefully!) in high school math or Calc 1. Every time you've seen an asymptote you've seen it from the side, where the curve diverges to +/- infinity; this is the most convenient way to show them when plotting functions of one (real) variable i.e. y = f(x) in 2d. Transfer functions are not functions of real numbers, though. They are functions of complex numbers (usually denoted s = σ + jω). In 2d it's more convenient to plot the function looking top-down and using the two axes (σ, jω) as the independent variables. The zeros (usually marked 'o') and poles (usually marked 'x') are then marked on this plane. 

You can think of the transfer function as a sheet located on/above this plane. Where there's a pole, the sheet is being propped up to infinity like a tent.  The value still blows up, but since there's two independent variables, it can be approached from any direction instead of the left or right which you're familiar with. Fundamentally you find the pole value the same way you did with real functions, except it can now be a complex number; it's not confined to an axis. 

How do poles affect the stability of a system?

Stability is a more involved topic. Poles are poles are poles and have less to do with stability than WHERE they are on the complex plane. You always want your poles to be on the left half of the complex plane. If you don't have a lot of experience with complex numbers, the most relevant detail here is that the jω term is oscillation (a sinusoid with frequency ω; mapped to the imaginary axis) and σ is the exponential growth factor which bounds the oscillation (real axis). On the left half of the complex plane, σ is negative and so your oscillations are damped to 0. On the right half, they'll grow to infinity. So you want the real component of a, Re(a) &lt; 0.

Poles and zeros are complementary singularities.  If you can move them to the same spot (like with a gain - but this is a more advanced topic) then they'll cancel each other out.