Audio equalization, more commonly referred to as EQ, is the process of altering the frequency response of an audio signal. The purpose of equalization is to increase or decrease the amplitude of chosen frequency components in the signal. This is achieved by applying one or more filters to the signal. 
There are a variety of reasons to apply EQ, and many producers consider it to be one of the most important audio effects. However, it takes a long time to get the hang of it, since there are so many possible ways to apply EQ. There are guidelines that can help producers avoid common mistakes and make good decisions about what frequencies to cut or boost, but ultimately practice makes perfect.
Cut before you boost. By subtracting frequencies to achieve the desire result, your mix will sound cleaner with fewer competing frequencies. Only boost if you cannot achieve the desired result otherwise.
Trust your ears. Quantitative guidelines, like "cut at X hertz to achieve Y sound" are important, but your ears should always have the final say. If you did something unconventional and it sounds good, keep it. If you followed the conventions but it doesn't sound good, try something else.
You can't polish a turd. EQ isn't going to be able to fix bad sounds. If you find yourself thinking that maybe EQ will fix up a bad sound, try replacing it with a better one. It could save you a lot of frustration and improve the quality of your output.
Tight boosts at the bottom. Remember, the human ear is more sensitive to frequency in the lower range than it is in the upper. When in doubt, use a tight (high Q) boosts at the bottom, and save wider boosts for the upper registers.
EQ in Practice
Parametric Equalizer Overview
A Parametric EQ is just a fancy name for an EQ that allows the user to choose different filter types, move the filters around to different frequencies, and specify either a Q-value or a gain level for each filter. Most parametric EQs include the following filter types to choose from. Each may be used alone or in combination with others to reach a desired effect.
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.
- Burg, Jennifer; Romney, Jason; Schwartz, Eric. "Digital Sound & Music: Concepts, Applications, and Science". Retrieved March 8, 2019.