I'm not a Welsh or Breton speaker (yet!), but my understanding is that Breton is a bit closer linguistically to Cornish than Welsh is. At least, that's what I've heard said, though I don't know either of our sister languages well enough to do any direct comparisons. I would guess that Cornish is sort of between the two and there are more differences between Breton and Welsh than Breton and Cornish or Welsh and Cornish. I think roughly speaking overall, there's up to 80% mutual comprehensibility between the three languages — as far as similar if not identical words go, at least. As Aran says, the mechanics of how those words are put together can be quite different.
Late 1700s — the last known community of a few fluent speakers was in Mousehole, near Penzance, in the 1770s — with some scanty knowledge of it (old sayings, poems, simple phrases, counting to 20 etc.) persisting among a few throughout the 1800s. The last known person with any significant traditional knowledge of Cornish, John Davey of Zennor, died in 1891 — we don't know exactly how much of the language he knew, but it's said he could hold a simple conversation in Cornish. (And that he kept his knowledge alive by speaking it to his cat. We're not told how the cat responded. )
Prior to that, though, including during the mid to late 1700s when there were still fluent native speakers alive, a number of scholars were collecting what information they could about the language and there were also some significant texts preserved from the language's heyday (late 1300s - 1600s), particularly plays on Biblical themes (the creation of the world, the Passion of Christ etc.) and lives of the saints. By the late 19th century there was increasing interest in reviving the language — along with a general revival of Celtic culture in the UK — and we generally date the real beginning of the revival from 1904 when Henry Jenner published the first textbook, A Handbook of the Cornish Language. I don't know all the details of how the language was reconstructed — I gather it's partly from the old texts, partly from comparisons with Welsh and Breton cognates, and sometimes just coining new words or new uses for old words when there's a need.
Anyway, in the past century and more it's gradually grown from a merely academic interest into a fully fledged modern language that one could easily live one's day-to-day life in if only there were enough committed speakers living close together in one community. There aren't yet, but you never know... Estimated numbers of Cornish speakers vary, but I think the general consensus is about 500-600 fluent speakers and several thousand more who have at least a little knowledge of the language. Which is a start. (I'm still a learner myself — I can hold a basic conversation and can make out the gist of a written text if it isn't too complicated.)