Pitch Shifting
Pitch Shifting alters the pitch of your playing.
There are two basic kinds: chromatic (or stepwise) and harmony (or intelligent, or sometimes called a harmonizer).
Chromatic simply shifts the pitch by some set amount, usually specified in terms of a chromatic interval (m2, M2, m3, M3, etc.), but often also allowing a continuous range of values in between the intervals.
A Harmony (or Intelligent) pitch shifter shifts the frequency by an interval of an unspecified quality (2nd, 3rd, 4th, etc.). It detects the pitch of the note being played and uses the corresponding interval in the scale selected. For instance, if you use the scale of C Major with a shift of a 3rd and play a C note, the shifter would output an E, a M3 above C. However, if you played an A, it would output a C which is only a m3 above A. Advanced intelligent shifters allow various scales beyond all the major and natural minor scales, such as harmonic minor (and some even allow you to program your own custom scales, including tuning outside of equal temperament).
Additionally, there are devices that are not dedicated pitch shifters, but make use of pitch shifting, such as octavers and some kinds of reverb/delay.
While uncommon to guitar, auto-tuners or pitch correctors are also pitch shifters that detect the pitch of an input and shift it to match the pitch of the closest note within a given scale. I will not cover them here.
Quality Considerations
Pitch shifting is more difficult than you may imagine. The basic principle is the same as changing the rotary speed of a turntable - by playing back a segment of audio more quickly, you increase the pitch; by playing it slower, you decrease pitch. For a device to do this continuously means it needs to buffer audio, leading to slight latency. A good pitch shifter should be low latency, feeling almost instantaneous. It also needs to piece these segments them back together to get a seamless audio output.
Harmonizers additionally have the difficulty of detecting the pitch you are playing, which has to occur before it knows by how much to shift the pitch. This analysis requires buffering and analyzing a segment of audio, which can lead to greater latency than a simple chromatic shift. Many chromatic shifts, however, also use pitch tracking to assist in reducing artifacts and smoothing output, which again can lead to lag.
Your guitar signal isn't a periodic waveform like a perfect sine wave, but real-world audio with changing waveforms, even when a single note is sustained without vibrato. Sped-up audio segments must be repeated to consume the same amount of time as their original length. Slowed-down segments must be chopped, so they don't consume more time than the original segment. Additionally, piecing together segments can get ugly - in order to align the phase, pitch must be detected...but what about those noisy transients?
Even when using "nice" intervals like octaves, the shifting process leaves lots of artifacts in the tone. If you have two shifters available, try shifting up an octave then back down to the original note. It will likely sound nothing like the original note. Using less harmonic intervals will fare even worse. A good pitch shifter minimizes artifacts, or makes the artifacts somewhat musical, rather than crackly or overly alien-sounding.
Advanced pitch shifters use a variety of methods to avoid or hide artifacts. They use multiple overlapping segments, adjusting phase from one to the next, avoiding creating "jumps" in the waveform, which would sound harsh and crackly. They also know how to intelligently extend a segment when pitch shifting upwards, again avoiding jumps in the waveform.
All this being said, some kinds of shifting artifacts might be desirable to some people or for some tones. Personally, I want to minimize them.
Common Features
Most pitch shifters feature at least 1 octave of shifting in either direction, usually offering 2, and some offering even more. Virtually all shifters offer a mix or volume control, to control the relative levels of the dry input signal and the pitch shifted note(s).
Some have multiple shifters built into one unit, so you can set multiple intervals above/below the input note(s) - so you could set the intervals to a 3rd and a 5th and keep the relative levels about even with the dry signal, producing a common triad from a single note input.
Advanced Features
Formant/Vowelizer
Some shifters allow you to target resonant peaks called formants. These are typically associated with the human voice, particularly vowels. The shape of the mouth used to make vowels forms resonant peaks, which gives each vowel its particular sound, regardless of its actual pitch (although it does have to have a fundamental pitch below the formant peaks). By applying such peaks to the pitch shifted tone, you get a more vowel-y sound, with the tone resembling the human voice among other instruments.
Pure Tuning
For a harmony shifter, some have the option of using pure tuning rather than equal temperament. See Equal Temperament and Alternate Tunings for more about the difference. Pure tuning is recommended - it will use the more harmonious ratios for the desired intervals, avoiding the beating that occurs when intervals slightly diverge from their pure interval ratio.
Special/User-Defined Scales
Simpler harmonizers only allow harmonizing against major and natural minor scales. More advanced ones will give greater scale options or even allow you to define a custom scale. This could be used for advanced/special scales such as harmonic minor, jazz minor, double harmonic, or Hungarian minor.
Some go even further, allowing you to specify scale notes in terms of cents away from their equal temperament tuning. This would allow you to harmonize in ancient Greek scales, for instance.
Polyphonic Shifting
As described above, reducing artifacts from shifting is not simple. Pitch tracking can help reduce them and smooth the output, but most pitch tracking is limited to monophonic (single pitch) tracking. Attempting to play chords while shifting can lead to greater instability of the shifted signal than without pitch tracking at all. In other words, it confuses the pitch tracking.
Polyphonic tracking can detect multiple pitches, retaining the benefits of pitch tracking when playing chords.
Sequencers
Some pitch effects may have a step sequencer, with different intervals programmed against a tempo.
Advice
Minimizing Artifacts
I tend to find pitch shifters' artifacts are less noticeable when shifting downwards rather than up. This is particularly true when mixing with the dry signal.
Placement
I tend to place chromatic shifts (usually used in the first two cases below) in front of my amp/distortion. The distortion tends to make the tone sound unified, like it's one instrument, not multiple guitars. It also tends to mask some of the artifacts.
Harmonizers are placed behind my amp/distortion. This gives the sound like there's multiple guitars playing harmonized lines, rather than one player playing intervals. With one guitar playing intervals into distortion, you get beating and intermodulation distortion that can take away from the fundamental pitches being played. It's fun and interesting to do on occasion, but when you're using a harmonizer, you tend to want to play lengthier harmonized lines, which will sound cleaner when mixed together post-distortion.
Uses
Transpose/Capo
The simplest use for a chromatic pitch shifter is to use it like a capo, but capable of lowering as well as raising the pitch of open position. This is useful if you want to play with friends or an impromptu gig where your guitar is tuned to standard E, but everyone else is tuned to standard Eb, or D, etc. Here you would use 100% mix, so you only hear the shifted signal.
Harmony Lines
A harmonizer is designed to play at a near equal dry/wet mix, so it sounds like two guitars playing in harmony. You can use 4ths or 5ths, but these are most often P4 and P5 intervals, at least in the common major/natural minor scales. More interesting is 3rds or 6ths. 2nds and 7ths can be a bit dissonant.
Usually, harmony lines are played like leads, in higher registers. This, combined with delay/reverb can mask any pitch shifting artifacts fairly well.
Changing the Overtone Structure of the Tone
Sometimes I like to use a chromatic shifter set to 4ths, 5ths, or octaves. This is similar to octaver pedals - it's not so much an attempt to bring other pitches into the mix, but to change the tone of the guitar. It can make the guitar sound kind of synth-like, as well as make it bigger and add some interesting overtones.
Whammy Pedal
If you can control the pitch of a chromatic shifter with an expression pedal, and it allows fine movement, not just movement in distinct chromatic intervals, you can set it up like a whammy pedal. This allows pitch motion similar to a whammy bar, but to a much greater degree. For instance, most whammy bars allow about a P5 of motion downwards and a M3 of motion upwards. A whammy pedal typically offers 2-3 octaves of motion in either direction.
12 String Tone
A simple octave shift mixed with the dry signal can give a 12-string-like sound. This can be expanded using multiple octaves.
Detuned
A slightly pitch-shifted signal mixed with the original signal gives a detune effect that sounds quite similar to a chorus. On some multi-fx units, this is a preferable mod sound to the actual chorus(es).
Digital Sound
The artifacts from pitch shifting can sometimes be desirable to create a "digital" sound, as if you are not actually playing guitar. Using high mix and multiple intervals, especially with a step sequencer or formant option can really mask the nature of the guitar. This is more emphasized when placed after distortion, but works even in front of distortion or on a clean tone.