In precipitation, radar signals are subject to signal attenuation - the simplest definition meaning it loses its range. Radar works by sending out a radio signal from the antenna. When it strikes an object, the portion of the signal that hits the object is reflected back to the antenna. Since the radar signal travels at a fixed speed (speed of light), it measures how long it took for the signal to go out and be returned, so it has the distance of the object. A speed-measuring radar uses Doppler principle - as the object moves closer, essentially the frequency of the return changes. The difference in frequency is measured and this gives a speed.
Now for a bit of somewhat irrelevant info:
Radar beams will bounce off anything, precipitation is one of them. As the precipitation gets heavier, more of the radar beam will be unable to penetrate the rain/snow/whatever and the effective range of the radar will be reduced. The size of the precipitation does matter - bigger rain drops will give a bigger return.
Weather radar actually measures how much of the beam is returned and at what distance to paint weather cells. When more of the signal is reflected back, it means more precipitation - therefore, stronger weather. The type of precipitation also matters. In terms of reflectivity, the strongest type of returns come from (in order of strongest to weakest) wet hail, rain, wet snow, dry hail, and then dry snow. So the radar will have a longer range in dry snow than in rain.
As for causing screwy things, if there was a substantial amount of wind and rain, the radar may pick up some Doppler shift of the rain drops, but that's about it. Weather Doppler radar is specifically built to look for it, but police radar isn't, and it doesn't have as much sensitivity. The only time I've known rainfall to cause weird things with radar equipment is if the rain got into the antenna, there was ice on the radome or the operator wasn't using the equipment properly.
