The notch filter is useful for blocking out heterodynes. If you tune in an AM signal on SSB and tune a little off frequency, you'll hear a loud single frequency tone. You can also hear this on AM when two stations key up together and they're a bit off frequency from each other (the frequency of the tone will be equal to the difference in frequency of the two carriers, so if one carrier is on 28.400.000 and the other is on 28.401.000, the difference is 1.000Khz, so you'll hear a 1000hz tone).
The notch filter produces a very narrow and very deep zone of attenuation, i.e. a notch in the receiver's passband. You can set the frequency range that's attenuated by adjusting the notch control knob. When you hear a single tone in the receiver that annoys you, turn on the notch filter switch and slowly sweep the notch control knob back and forth a few times. When you get the notch right on the same frequency as the tone, it should disappear (or at least be drastically reduced). This can be a little tricky depending on the shape of the filter and the coarseness of the control.
This does have an effect on the audio of the person you're trying to listen to because you're "notching out" a part of their audio envelope too, but if the filter is designed well, the notch should be narrow enough to reduce or eliminate the unwanted tone without hurting the intelligibility of the other guy's audio too badly. What you sacrifice in audio quality may be worth it to get that damn screeching out of your ears.
Some radios have audio notch filters while some have IF-level notch filters instead. And some have automatic heterodyne blockers that you don't even need to tune manually. (You can also buy or make an external audio-level notch filter for rigs that don't have them built in.)
The notch filter is most useful for tones that are close to the middle of your passband. Unwanted signals at the edges of the passband can often be masked using the IF shift control, or by selecting a narrower filter (if your rig has one).
Basically, the radio gives you the tools, and you get to pick the best one for the job.
-Bill