There are three main parts to a thumb drive. There’s a chip that interfaces with the USB port so that they can “talk” to each other; an optional controller chip to help manage the reads/writes; and the flash memory chip(s) to store the data. The first two may be combined into one chip sometimes.
The flash memory stores the data in the form of electrical charges. They use a special type of transistor that includes a “floating gate”…a part of the transistor that is not electrically connected to anything. Charges can be pushed onto or pulled off of the floating gate through the insulator around it because it is very thin and a relatively large voltage is used.
By putting charge on the gate or not, each flash memory transistor can store a 1 or a 0. It’s even possible to put more than just two values into a single transistor by using varying amounts of charge. This improves density/cost, but hurts reliability.
The flash controller does various things. Probably the most important is to do various tricks to try to manage the lifetime of the memory devices. They eventually wear out, so it’s important to avoid repeatedly reading/writing the same transistor. Many thumb drives do this poorly or not at all.
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