Cooties are lice. The origin in English is a little unclear. I had believed it was borrowed from Māori or Malay or something ('kutu' is the almost universal word for 'louse' across Austronesian languages), but the Oxford English dictionary explicitly disagrees. They believe it comes from a reputation the coot (the bird) had for having a lot of parasites, and cite a saying going back to the 1860s that something is as 'lousy as a coot'.

Anyway, it entered English during the First World War as soldiers slang, the OED says originally as the adjective 'cooty' ("Two of the four corporals have celebrated the occasion by ‘going cooty’, otherwise declaring possession of one or more lice" from April 1915), then the noun 'coot' ("Willet very pale and excited grappling with an enormous ‘coot’ (otherwise louse).", same guy, September 1915), then the plural 'cooties' by 1917.

Cooties as imaginary contagion of some sort is American children's slang from the 1950s or 60s. The OED has two quotations from the 50s, but they point out that both may be transitional and refer to actual lice, and then a 1967 book by Beverly Cleary where the 'cooties' are clearly no longer lice, and can be spread through breathing. Bear in mind of course that children's slang will always have a bit of a lag before appearing in print.

The Oxford English Dictionary isn't free (I read it through my library), but you can check the other OED (the 'Online Etymology Dictionary') and Wiktionary if you want, both of which only give the Malay/Māori origin. I'm going to say though, I don't find that convincing any more (it was the 'lousy as a coot' thing that convinced me).

There's also a disagreement about current usage. Wiktionary says the old meaning is 'dated soldiers slang', while the OED says the new meaning is 'chiefly U.S'. I know to my mother (UK or Ireland) cooties are still lice, but the American meaning might be taking over.