A high pass filter sets a down point and reduces the volume of any frequencies below that point. Generally, the down point is the frequency where a -3dB gain is applied (see example below, which shows a -3dB down point at 1kHz).  The amount of reduction below the down point depends on the shape of the attenuation curve. Some EQs allow the user to choose from a variety of different curves, which are usually measured in dB/octave.
Higher dB/octave low pass filters (e.g. 48dB/octave) are better for completely eliminating frequencies below the down point, and more gradual curve (e.g. 12dB/octave) will provide a less drastic effect.
Aside from the down point, high pass filters also allow the user to adjust the quality factor of the filter, commonly known as Q. If Q is greater than 0.707, the filter will introduce resonance, and if it is less than 0.707, the roll off at the down point will be greater. 
Filter resonance is when frequencies are boosted at the down point, right before the filter starts cutting them off. In electronic music, filter resonance is crucial for giving character to synths and other sounds. Another application of filter resonance is to automate the down point over time, which can give the underlying sound a 'wah'-like effect that has become commonplace across many genres of electronic music. Note: this is more commonly used in combination with low-pass filters. [citation]
If you are producing music that emphasized the bass and kick drum, try using high pass filters on your other instruments to clear out plenty of space in the low end for the kick and bass. Some producers will even put a high pass filter on every sound other than the kick and bass. [citation]
A low pass filter has the same mechanics as a high pass filter, except for that it attenuate frequencies above a certain threshold (the down point).
A low shelf filter allows all frequencies to pass through, but cuts or boosts all frequencies below a certain threshold. In addition to a Q value, the filter uses a gain value. Negative = cut; positive gain = boost.
A low shelf is good for controlling the overall low end of a sound, as opposed to making a 'surgical' cut or boost at a certain frequency. For example, if you want to reduce the low end of a piano to make the bass stand out, a low shelf cut might be a good idea. You can reduce the low frequencies of the piano without eliminating the entire low end. [citation]
Using low/high shelf instead of a low/high pass is a more subtle way to EQ your sounds, which may be all that is necessary if you are sounds that are already of high quality. [citation]
A high shelf is identical to a low shelf except that it applies a cut or boost to all frequencies above a threshold. It is good for controlling the overall high end of a sound.
The bell filter (sometimes called peak-notch) is used to boost or cut certain frequency ranges centered around a certain point. In addition to a Q value, it has a gain setting that determines how much volume is added or removed around the target frequency.
It is most often used to subtlety change the character of a sound, eliminate certain unwanted frequencies, accentuate certain desirable frequencies, or it may be used in conjunction with frequency automation to provide a sense of movement over the length of a sound.
Notch filters completely eliminate a certain frequency, and reduce the surrounding frequencies according to the Q value that is set in the filter.
Use notch filters sparingly, as they can have a large impact on the underlying sound and can introduce unwanted side effects.