Its not possible to list all functions here, they are too specific for each tool. But we can describe families of functions and their impact on the music.

Available since the tape loop phase, 1960:

Overdub

Set to record into the loop, switched off to solo over the playing loop.
Technically, this feature can be added to any delay by connecting a volume pedal at its input (an mixing the dry signal to its output).
Philosophically it means to choose between building a base or enjoying it, between evolvement and contemplation, between stepping and preparing for the next step.
Musically each overdub means to hire another musician and teach him on the spot what he has to play. Once he starts playing that, its not easy to stop him, though 🙂

Feedback

This is the oldest and probably deepest and hardest to understand parameter.
Its the volume of the present repetition appearing in the next repetition.
In other words:
if closed, all is repeated only once,
if fully open, it repeats infinitely (in the case of Delay Looping a technically limited amount, with increasing noise) and
in the mid positions the loop fades over any period. [ Its also possible to amplify (more than fully open), so the loop grows, but this has not been of use and therefor not implemented in dedicated looping tools. ]
Technically, this feature can be added to any delay with a volume pedal that runs from the output to a mixer at the input. Its the oldest function and still the most natural way to let the loop evolve: While the old fades, there is space for new, it becomes possible to draw over the picture where desired, even if its just one note, or perfection the whole loop while its fading. Some play a phrase over and over and let the rest of the loop fade, or let it fade while playing over it and finding a different idea that they finally overdub and such take the loop into a different direction.
Unless Feedback is closed and all replaced, theoretically all that ever was played into the loop is still there, although often inaudible. Since we heard a part louder in a earlier part, we can easier perceive it when almost faded out in the later part. So this musical structure allows to be aware of much more musical figures at once than in any other structure!
Dedicated looper tools sometimes do not have this function at all, or only in steps, in certain functions.
Philosophically, its the amount of attachment to the past, the speed of development, the presence of the tradition,,,
By system theory, Feedback is present in any natural process. Its the natures way to correct inaccuracy, and (if feedback bigger than one/full) to create oscillations.
Musically its the cure to bore. Whenever we become saturated of the loop, we should reduce Feedback and let us inspire to play something new into the loop.

Speed change

Changes length of the loop and pitch at the same time. Not easy to use!
Technically naturally available for the tape recorders and easy to offer for the older digital delays by changing the sample clock, but more complicated for the digitally processing loop tools. The tape speed could not be changed very quickly and usually was not accurate. The older digital processors only offered half speed, but immediately. Newer loopers allow precise harmonic steps, controllable by a bass pedal, for example.
Musically, there are strongly different uses: The tape speed change was limited to some yowling or slow changes. The half speed function is pretty easy to use, because it doubles the rhythm and changes by one octave, so there is a continuity, but still a drastic change. It allows for example to record a bass line as a double speed melody and then drop it down. Starting at the lower speed, incredibly quick music can be created. Newer tools can be controlled with a MIDI keyboard and if the loop is a single smooth note or sound wash, the time change is not noticed and a melody can be played with the loop. If only one of several parallel loops (multitracking) is changed, the change becomes less shocking and specialists like Per Boysen manage to consciously work with the arising polyrhythms and harmonic changes!

Mute

Technically simply shuts off the loops output.
Musically interesting to create breaks. To allow breaks of free length, dedicated tools generally allow to play the loop from the beginning when the break is ended.

Available since the Echoplex phase, 1992:

Record

The first hit starts to record the first loop and the second hit defines the loop time and starts repeating. A silent short thow foot switch is needed and usually some practice to achieve precise rhythm.
One might think this is the most basic oldest function, but it was first introduced with the Paradis LOOP delay in 1992! Before, during 30 years, livelooping music was created by defining the loop time first and then playing into it, which made it difficult to create an accurate rhythm. A click track or synced drum machine can help, but often, an introduction without loop is played before Record, so its important that the timing is given by the musician, instead of by the machine.
Philosophically it means to define what you more or less will have to live with for the present trip. You can change a lot, but timing and pitch only with newer tools and some difficulties.
But its also helpful for the listener to hear this definition, it creates confidence…

Replace

As the loop plays, it’s possible to replace sections of it. It’s a bit like “punching in” on a tape deck, except that a high degree of rhythmic accuracy is possible. Skilled players can quickly create glitchy rhythms using this technique.

Some devices will allow replace which is automatically confined to a subdivision of the loop time. If this is used with sustaining sounds it’s easy to produce rhythmic pulses and sequences with perfect timing.

Multiply-Divide

Its easier and quicker to start a short loop and at some point extend it to bring variations or chord changes into it. To maintain the precise repetition of the short loops content, the tool rounds the extension to an exact multiple of the original.
Its also possible to crop the loop to allow quicker changes, or if not rounded to a multiple of the original loop length, cropping creates interesting new rhythms. Usually its one loop length or since the startpoint of the loop, some tools even erase all overdub and only leave the very first loop.

Undo

This function was created because it was painful to destroy a loop with a wrong note. Just like in any computer program, Undo erases the last action, so the wrong note is never repeated. In praxis its not quite so clear how much should be erased.
Philosophically, this is the least natural function so far. Its not possible to make a creation disappear completely, but since we got used to trash cans which content then simply disappear with a truck… its an equally relaxing and dangerous function in that it seduces to try things more creatively and less responsibly.

Redo

For those who dont know what they want, one might think. But its also the most simple way to

Reverse

A loop can run backwards. There are many options to do so, but most tools run backward from the point Reverse is pressed. So if not hit exactly, the rhythm is lost. Melodies are inverted, which can be very interesting, but also each note runs backwards which changes the sound drastically. Some users like this sound more than the melody inversion. Especially interesting when used quantized (at loop start) on one of several tracks.

Retrigger

The loop is immediately played from its beginning again. Can be used to synchronize to the drifting band or fellow loop, manually or by automatically waiting for the next sync signal (MIDI…)
Retrigger can also be used for stutter effects as they are known from the DJs.

Sync

The fundamental function to play together. One of the major motivation to create the LOOP delay, yet little used by the musicians.
Its important to understand that an initial sync of the loop length is not enough to keep it forever. The master needs to send the sync signal continuously and the slave needs to be able to adapt to it whenever the timing difference becomes unacceptable.
MIDI allows to sync the looper to a drum machine or sequencer or another looper, being that the direction (Master-Slave) has to be defined with cables and settings.

Available since the Repeater phase, 2001:

Tempo/Pitch change

Sophisticated tools allow to change tempo and pitch independently, which is a lot more musical than the simple speed change. Depending on the quality of the algorithm, the sound deteriorates between almost inaudible and ugly.

Multitracking

Instead of using several loop units, a single one can record loops in parallel so the user can fade parts of the loop in and out and mix them with separat effects and such.
Depending on the unit, the loops can be of different length and in sync or not, so polyrhythms and more complex structures can be created.

[please add newer functions here!]

Matthias Grob, 2009
additions by Michael Peters