There has been much debate on what equipment Boards of Canada uses to get their unique sound. Due to their reclusive nature and lack of touring, not much is known for certain, but there is much speculation.
BoC themselves have, in multiple interviews, mentioned the use of a "secret weapon" they use to generate their unique sound. As the name implied, they have declined to provide any specific information on what this "weapon" might be, but most expect it to be some sort of old analog synth.
Many believe the Yamaha CS80 to be this weapon. In the WATMM thread "ATTN BOC-sessives, CS-80 on eBay", however, jbible refutes this, saying
"The synth used by BOC often mistaken for a CS80 is a Crumar with a BOC emblem covering the name of the synth on the back."
Since jbible has seen BoC live (and is the source of the Live @ ATP bootleg), his comments hold some weight.
Here is a detailed "write-up" of the CS-80, courtesy of the SynthMuseum.
In the original album description provided by Boards of Canada's label (Warp Records) the Yamaha CS-70M is mentioned as their "trademark" synth. 
It appears that BOC use the SH-101 extensively. It possible that it is their only mono-synth.
It's hard to prove this definitively and the tone of the sounds are nearly always altered in the mix making a perfect copy of any particular sound extremely difficult.
However, having used this synth extensively in my opinion there's almost no mono-synth sound on BOC's records that couldn't have come from the 101.
In this link I show how to make the Roygbiv bassline on a 101 (actulaly a 202 but the sound generation circuitry is identical). WATMM forum link
A blue Yamaha CS-1x was mistakenly visible in a 1999 Lighthouse Party photograph. Members on Twoism and WATMM later recognized the blue synth as the Yamaha AN1x (see below).
The Yamaha AN1x has been recognized as being the blue keyboard in question. twoism.org have a thread on this, where some members have analysed the photograph, and by altering contrast etc, have stated that certain ports that only the CS1x has, are 'missing' from the back panel, thus strongly suggesting that the synth in question is indeed an AN1x. Interestingly, in recent years, numerous examples of AN1x programming to emulate/re-create BOC sounds have surfaced, demonstrating remarkable closeness to BOC, thus furthering the potential that the blue synth is in fact an AN1x. Note that BOC only used the AN1x to trigger live samples. 
In a 2001(?) interview with Mate Galic of Native Instruments, Boards of Canada is listed as one of the artists using NI products (but not details on which ones). 
“We love these low-quality tape machines,” Eoin says. “The great thing with machines such as the Grundig is that it's tragically bad. Whatever you record into it just doesn't come out unscathed. There's a ‘magic eye’ valve display on it, and when you hit the tape deck with the right volume, enough to fill out the magic eye, it's at that exact sweet spot that it is saturating the tape. So if you then sample back the playback, it's got a thousand years' grain on it.” Remix, 2005
We drop a lot of our music down onto a Tascam 4-track that has a great saturating effect on the sound. Remix, 2002
Whether they're working separately or together, getting ideas down is generally a result of recording extended jams to tape on anything from a Tascam MSR-16 reel-to-reel to an old Revox recorder to a Grundig machine to an ordinary cassette. Remix, 2005
Based on the photos from Warp10, the Lighthouse, and ATP, the mixer BoC used was the Soundcraft Spirit Folio SX. Since BoC ran everything through this live, they were running their samplers/synths through these Folio pre's, meaning that a noticeable part of BoC's sound is from this mixer. Assuming they used the Folio when recording the Peel Session, and given the quality of those songs and the soundboard quality of Julie and Candy, it's safe to assume that BoC never really owned any high-end rackmount preamps (Neve 1073's, LA 610's, API 512c, etc.).
They most likely ran a few rackmount effects into the aux channels of the Folio, probably to get their phaser/reverb sounds, but so far none of those units have been deduced from available photos.
The rear of an Yamaha A3000 can be seen in this live photograph: 
We have five or six samplers, but my favorite by far is still the Akai S1000. It's an old tank now, and the screen has faded so that I almost can't read it, but I know it inside out. It's the most spontaneous thing for making up little tunes. It adds something to the sound — maybe the lower bit depth has something to do with that. - Remix, 2002
The rear of an Akai S3200 can be seen in these live photographs:  
As for our percussion, it's never just a drum machine or a sample, we put a lot of real live drumming or percussion in there, woven into the rhythm tracks, and it brings a bit of chaos into the sound that you just can't achieve any other way