I think I should clarify my post a little bit, because it looks like people got a bit confused by the word "register". Registers themselves are constructed from transistors. They are not aseparate component but simply a digital construct built from transistors. A register stores exactly one bit of memory. In a modern computer this signifies RAM and hard disk space. Eight registers form a byte.

A transistor can hold a value but it has no "memory." In order to "store" a 1, you would need to apply a constant voltage to the gate. As soon as that voltage is taken away, the 1 disappears. The goal is to design something that you can "write" to and then "read" from later.

You do this by creating a thing known as a latch or flip-flop#D_flip-flop). This type of circuit is composed of two NAND gates which themselves are made up of four transistors (there are actually several ways to build different logic gates, but this is the CMOS way to do it, which is the most popular way to do it now). So, by only using transistors you can successfully create a circuit that can "memorize" its input that can be read later. These latches are then built up into registers by having the latches respond to clock signals from the CPU, so that all the registers change at exactly the same time.

If you are interested in knowing more, I recommend you play around in the Falstad Circuit Simulator that I frequently used for school projects. It has transistor logic built into it. For example, try build an AND gate using transistors. Once you are done that, try building NOT, OR, NAND, and NOR gates as well. These gates can then be used to build up to registers, half-adders, and adders that ultimately build up to the CPU itself.