Semi-conductors are materials in which nearly all the electrons are paired off and inert, but a very few electrons are sailing for the New World or taking jobs as pirates on the open sea. The exact impurities ("dopants") added to the original material (which is often but not always a crystal) affect the band gap that inhibits an electron from bolting. When semi-conductor materials are combined into the form we call a transistor, its band gap controls the precise amount of light or electric current needed to goose the transistor into its "on" position. ("Transistor" is a special form of semi-conductor that's sandwiched and wired up a particular way, but the terms are linguistically related: a transistor neither transfers nor resists indiscriminately, just as a semi-conductor neither conducts nor insulates completely. Instead, like an efficient doorman, they both let through just the electrons we want to pass.)
Tiny electrical-current devices that can be efficiently switched on and off with tiny amounts of electricity lend themselves to compact logic circuits. Transistors can be hooked up so that their output connections feed back into their inputs, an arrangement called a "logic gate." A transistor in one of these arrangements stays on even when the base current is removed, but when a new base current flows, the transistor flips back off, then on again with a new current, and so on. This is called a flip-flop, which amounts to a simple memory device that stores a zero (when it's off) or a one (when it's on). It is the basic technology behind computer memory chips. They are simple or complex depending on our ingenuity in constructing the interactive logic gates.
Modern, miniaturized logic circuits are laid down on a chip by a kind of etching process. For instance, a tiny little light pattern can be shone on the chip, and then a circuit material is painted on in an incredibly thin coat that sticks differently depending on where the light hit the surface. Chip-makers have gotten so good at this miniaturization that they're approaching the nano-scale--still bigger than an individual atom, but getting near that neighborhood. The smaller the wavelength of the light, the finer the pattern we can achieve. Visible light is in the 400-700 nanometer range, but of course wavelengths get smaller and smaller as you move up into the ultraviolet and gamma-ray ranges. If we can get the etching pattern down to the atomic scale (1/10 of a nanometer), we'll obviously be able to pack a lot more circuits into a small space.
I think I always had the notion that silicon chips were made of the same material as beach sand. Sand is really silicone dioxide, though, whereas the silicon in chips is elemental, crystallized silicon, which looks a bit like a silvery metal.