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.
"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."
Here is a detailed "write-up" of the CS-80, courtesy of the SynthMuseum.
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 is possibly visible in a 1999 Lighthouse Party photograph. If it is a CS1x, it is more likely to have been used to trigger samples than generating its own sound, though this could ostensibly have been used for such sounds as Olson's filtered piano.
(Special Note): Yamaha's AN1x has also been quoted 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 a certain port that only the CS1x has, is '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. 
“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
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
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