Pulsars are rapidly-rotating stellar remnants that we see as pulsing radio or X-ray sources.
That is, once a star has burned all its fuel, there is no longer heat available from fusion to keep its tremendous mass from collapsing inward due to gravity. (Usually there is a supernova that burns up the last of the fuel, and that can blow off some or all of the star's matter. I'm going to ignore that for the moment and look at what happens to what's left.) If there's not too much gas left, as it cools and contracts, eventually it will be supported by the reluctance of electrons to be in the same place at the same time. If the remnant is more massive than that, the electrons can't support it and the star collapses further, until it is supported by the reluctance of neutrons to be in the same place at the same time. If the remnant is even more massive, nothing can support it, and it collapses into a black hole.
Pulsars are the middle case: stellar remnants crushed into a giant ball of (mostly) neutrons. When you crush a star this much, since it keeps most of its angular momentum, it begins rotating rapidly (pulsars are observed to rotate with periods from about 1.4 milliseconds to about 8.5 seconds).
This crushing (and possibly some other processes) also raises the magnetic field density up to about 10^12 Gauss. This is one of those astronomical numbers that's so large it's hard to make sense of; it's a hundred million times stronger - more dense - than the modern "supermagnets"; the energy density of the magnetic field is so high that using E=mc^2 it has a mass density about that of lead - with no matter at all in it.
Not every neutron star is a pulsar, though; we call them pulsars only when we see pulsating radiation coming from them. This "radiation" can be radio waves (which is how the first pulsars were found), coming from some poorly-understood process that's maybe a little like the Earth's aurora. Like the Earth's aurora, there is plasma - loose electrons and ions (and actually, in pulsars, a lot of positrons) - and it is channelled along field lines. Near the magnetic poles this plasma is densest and most active, and it emits directional radio waves (as the Earth's aurora does on a vastly weaker scale). These beams of radio waves sweep around the sky as the pulsar rotates, and when they pass the Earth, we see a pulse of radio waves.
The radiation can also be higher-frequency radiation: many pulsars emit X-rays, either from a hot surface or from the magnetosphere (the region of strong magnetic field above the surface), and the new Fermi satellite is finding that many pulsars actually emit gamma rays (even higher-frequency photons). A few pulsars even emit detectable optical pulsations, though sadly I don't know of any that are both bright enough to see through a modest telescope and slow enough to visibly blink.
Pulsars are interesting to astronomers for any number of reasons, but one of the most valuable is that they serve as clocks: since their pulses come from rotation, the pulses come in very regularly. To make the pulses drift or wander, the way my wristwatch does, some process would need to apply a tremendous torque to a very massive body. Not many processes are available to do this, and observationally, pulsars spin with astonishing regularity. The only reason we are able to distinguish any deviations from regularity (a topic for another post) is because we are exceedingly good at measuring time.
Having these clocks has proved extremely valuable: when a pulsar and another star are orbiting each other, we can measure the orbit extremely accuately. Hulse and Taylor won a Nobel prize for measuring the loss of orbital energy in one such system due to gravitational waves; a more recent project aims to observe gravitational waves from elsewhere in the universe by correlating pulse arrival times from many pulsars. Pulsar timing allows us to measure very accurate positions, distances, and proper motions to some pulsars. So the study of pulsars has been extremely fruitful - even though, forty years after their discovery, we still don't really understand how they produce the emission that we see.