The AKAI Headrush Pedal and the Boomerang III
Mal's Loop Recorders of choice.

The Akai Headrush E1 pedal is the looper (sample loop recording pedal) I mostly use (I called it Derek, but I also have another called Dolly). It's great for those into simplicity and spontaneity. It does 11 seconds with infinite layers or 22 seconds of just one layer. It has excellent sound and it has an inputting advantage over most other loopers in that when recording a loop, you can can just keep stomping the start button until you're happy with a start point. This makes it great for impro sessions. And once you're adding layers, you can always can cut back to the initial loop. Akai have stopped making the E1 and replaced it with Headrush E2, which is pretty much the same except for 6 seconds extra loop length and a lovely blue hue. Yay! Mr Percival and Adam Page are also fans of the Headrush.

Boomerang III
  I also use the fabulous Boomerang III . Great compact size, up to 4 separate loops (synched or unsynched), long loop times, undo, infinite layers, reverse, half time, copying sideways. It's about $500 if you buy it from the maker, Mike (who is in Texas). There's a few little things about it that could be improved (with a software update) that would make it AMAZING rather than just bloody good... I'm lobbying Mike about it, so stay tuned. While I'll always use a Headrush too, I highly recommend the Rang.

Other Loopers
I've also got a Korg Kaos Pad KP3, which is another beast all together (effects and looping). It's a lot of fun, but as a looper, it's very tempo based and thus a little restrictive. And it's a bit tricky to use hands free! This is what Beardyman uses.
And I've also had a Digitech Jamman, which is one loop at a time (with layers), but it lets you save up to 99 loops and has a USB for loading samples from elsewhere. Classy.
Other loopers include: The Boss Loop Stations (RC2, RC20, RC30, RC50, RC300, RC505) which have lots of good features (the bigger the number, the more the features). Early Boss Loop Stations had rubbish sound, but that's improved over time. The RC50 is a fine 3 loop beast and is definitely the industry standard. But it's huge and has some annoying features too. And, I'll be honest, I find the Roland/Boss market dominance irritating. There's just a few key things the RC50 can do that the Boomerang can't (and vice versa... bring on that Rang software update, I say!) Mihirangi, Adam Page and many others use  the RC50. Linsey Pollak uses an RC300 (he won it in a competition!) It's like 3 RC20s and an effect unit all combined and well integrated: If you like the Boss gear, it is rather fab, although many people prefer the RC50 layout: It's what you're used to I guess. The RC505 is their latest offering, with 5 phrases (synched or not) and computer interface, but not so good if you need your hands for other things, although I have friends who are using it with bare feet and it works for them!
The Ditto looper by TC Electronics is nice and simple and well laid out.
There's also the Line 6 DL4 (good inputting, 16 sec max loop, nice green... Reggie Watts uses one of these), the Gibson Echoplex (a rackmount unit that does everything and is very flexible and adaptable, but thus very complicated), the VoiceLive touch (a strange beast, which doesn't grab me, but it does do auto harmonies!) and the Electro-Harmonix 2880 Super Looper (really fab, but not so good hands-free). There's also the Electro Harmonix 720, which Chinese company Ammoon do an perfect knock off of: It's a stereo looper which has excellent sound, a 10 loop memory, undo, reverse, fade out and seemless loops (no glitch on the start). Ammoon also make an even cheaper one button looper which is damn fine too.

Software Loopers
I've mucked around a little with softwear loopers, like Abelton Live and Fruityloops (now FL Studio), but I don't like to have a computer onstage. And there's the brilliant Everyday Looper app for iPhone/iPad, but that's tricky to get sound into and not so good hands free! When recording for CDs, it's better to use a proper recording software, so you have more control (for panning, EQ, reverb, etc). I use a software called Reaper, which is a brilliant, flexible and cheap to boot! . When recording someone else looping, I find it best to let them do their thing, but record a direct channel as well, so you can recreate their looping later (edit fest!)

Loop recorders do great things for your timing, as you have to be very accurate with the buttons... it's like any instrument: You have to play it well. I only say this because I know of many people who were extremely frustrated when they first tried it and things weren't as easy as they seemed to be. It just takes practice.

Other Tech Stuff
On the sound front, I use a little mixer on stage to control my inputs. I send the left output to the PA and right output to my loopers, from which I return to an aux channel (which is panned left). It means that I can use the pan on each input channel to choose how much of each signal, if any, goes to the loopers... that set up took me a while to figure out! I use a Behringer 1002fx, which is a fine and tiny mixer.
I run my loopers through a switch I made that allows me to choose the order of the loopers in series. I call it the Hexapus:

There are two vocal mics I currently use live: A Shure Beta 57A and a Sennheiser 835, depending on the situation. Both mics have good bottom end for beatboxing (but thus require CCMP when speaking or singing). The Beta 57A has a more high mids and is smaller, which is good for different cupping, etc. But the 835 is more solidly built (less rattle), which is good for mic flapping/face bashing... although it is heavier... it's a trade off! On my bubblewrap stomp and mbira, I use the capsule out of a Behringer XM8500, which, while amazingly cheap, has surprizingly good bass response... I used one of these as my vocal mic for a long time.

If you don't use a mixer and your looper only has jack input, always make sure you convert the impedance! Here, I'll explain: Impedance is a bit like resistance (but the AC equivalent) and if at any point you're going from XLR (the three pin mic plug, aka canon, which is low impedance) to jack (guitar plug, which is high impedance), a lead or adaptor that just has XLR at one end and jack at the other isn't good enough: There needs to be a little circuit in there to convert the impedance. You can get them from good guitar/electronics shops, but beware of any gormless 23 year olds (or know-it-all 40 somethings) who tell you it doesn't exist or it's not necessary: It does make a difference to the sound.

Good luck! Cyasoon, thine a looping, Mal