Kodak started making "pocket" cameras in 1895, and each new design used a different film format — different sizes, aspect ratios, cartridge types, and orientations. By 1908, they decided to simplify the confusion with a numbering scheme, calling that first format "101" and continuing the numbering up from there. So, that's the scheme — the numbers correspond to the order early Kodak cameras were released, starting with 101.
The 120 medium-format film still in use today is part of this sequence. However, when 35mm film came out, it appears they skipped from 130 ahead to 135, presumably for the nice alignment. And by the 1930s, they'd introduced some other variations — 620 and 616 were the same as 120 and 116 but with smaller spool diameters, and 220 was the same as but twice the length of 120.
Then, in the 1960s, Instamatic cameras were given 126 film, since the image size is about 26mm square — this matches the 135 and 35mm alignment, and the original 126 was no longer in use. And then when the smaller "Pocket Instamatic" cameras were introduced, they used 110 (again, discontinued in the original series long ago), apparently because it's smaller than 126 and "one-ten" was nicer to say than the possibly-logical 113 or 117.
Film for APS-C is 240 — I guess because, hey, why not.
I actually get into this a little bit in my answer to What historic reasons are there for common aspect ratios?, although obviously this is not a duplicate question. More on the numbering history can be found at these web pages: History of Kodak Roll Film Numbers and The History of Kodak Roll Films.