A pole is a frequency where a filter resonates and would, at least mathematically, have infinite gain. A zero is where it blocks a frequency - zero gain.
A simple DC blocking capacitor, such as for coupling audio amplifiers, has a zero at the origin - it blocks 0Hz signals, that is, blocks constant voltage.
Generally, we're dealing with complex frequencies. We consider not just signals that are sums of sine/cosine waves, like Fourier did; we theorize about exponentially growing or decaying sines/cosines. Poles and zeros representing such signals can be anywhere in the complex plane.
If a pole is close to the real axis, which represents normal steady sine waves, that represents a sharply tuned bandpass filter, like a high quality LC circuit. If it's far, it's a mushy soft bandpass filter with a low 'Q' value. The same kind of intuitive reasoning applies to zeros - sharper notches in the response spectrum occur where zeros are close to the real axis.
The transfer function L(s) describing a filter's response should have equal numbers of poles and zeros. This is a basic fact in complex analysis, valid because we're dealing with linear lumped components described by simple algebra, derivatives and integrals, and we can describe sines/cosines as complex exponential functions. This kind of math is analytic everywhere. It is common to not mention poles or zeros at infinity, however.
Either entity, if not on the real axis, will appear in pairs - at a complex frequency and at its complex conjugate. This relates to the fact that a real signals in result in real signals out. We don't measure complex number voltages. (Things get more interesting in the microwave world.)
If L(s)= 1/s, that is a pole at the origin and a zero at infinity. This is the function for an integrator. Apply a constant voltage, and the gain is infinity - the output climbs without limit (until it reaches the supply voltage or the ciruit smokes). At the opposite end, putting a very high frequency into an integrator won't have any effect; it gets averaged out to zero over time.
Poles in the "right half plane" represents a resonance at some frequency that makes a signal grow exponentially. So you want poles in the left half plane, meaning that for any arbitrary signal put into the filter, the output will ultimately decay to zero. That's for a normal filter. Of course, oscillators are supposed to oscillate. They maintain a steady signal due to nonlinearities - transistors can't put out more than Vcc or less than 0 volts for output.
When you look at a frequency response plot, you might guess that every bump corresponds to a pole, and every dip to a zero, but that's not strictly true. and poles and zeros far from the real axis have effects that aren't apparent in that way. It would be nice if someone invented a Flash or java web applet that let you move several poles and zeros around anywhere, and plot the response.
All this is oversimplified, but should give some intuitive idea about what poles and zeros mean.