Trainbands were companies of militia in England or the Americas, first organized in the 16th century and dissolved in the 18th. The term was used after this time to describe the London militia. In the early American colonies the trainband was the most basic tactical unit. However, no standard company size ever existed and variations were wide. As population grew these companies were organized into regiments to allow better management. But trainbands were not combat units. Generally, upon reaching a certain age a man was required to join the local trainband in which he received periodic training for the next couple of decades. In wartime military forces were formed by selecting men from trainbands on an individual basis and then forming them into a fighting unit.
The exact derivation and usage is not clear. A nineteenth-century dictionary says, under "Train":
"train-band, i.e. train'd band, a band of trained men, Cowper, Joh Gilpin, st. I, and used by Dryden and Clarendon (Todd)"
— Skeat's Etymological Dictionary of the English Language (Oxford 1879)
The issue is whether the men "received training" in the modern sense, or whether they were "in the train" or retinue or were otherwise organized around a military "train" as in horse-drawn artillery.