Voltage is easy to measure using a voltmeter or multimeter. Current is a bit trickier, but not too difficult in “normal” ranges if you know what you’re doing. If, however, you want to measure a tiny amount of current, things become more complicated.
For taking tiny measurements, for instance to measure how much current a sleeping microcontroller uses, David Johnson-Davies of Technoblogy has developed a Nano Current Meter capable of measuring between 10µA and 30nA "with reasonable accuracy.” Put in different terms (as µA means microamps, or 10^-6 amps, and nA means nanoamps or 10^-9 amps) the device can measure between .00001A and .00000003A—very, very small numbers indeed.
As Johnson-Davies explains, measuring such small currents with a voltmeter is generally impractical with a normal voltmeter, or requires expensive additional equipment. What he came up with is a simple current meter circuit based on capacitor discharge. It uses an ATtiny84 to measure the voltage across a capacitor, and thus infer the current drawn by this change, and a 7-segment display provides the needed user output.
One interesting effect of this type of circuit is that at very low current usages it can take more than a minute to discharge the capacitor. Even so, the project would be extremely useful in many situations, and using a capacitor discharge in this manner particularly clever. Be sure to check out more details on the build via his project write-up.