What a micrometer is
A micrometer (micrometre outside the US) is one millionth of a meter, written μm: the Greek letter mu μ for the prefix micro, plus m for meter. There are 1,000 μm in a millimeter and 1,000 nanometers in a micrometer.
It is the natural unit for very small things. A red blood cell is about 7 to 8 μm across, many bacteria are 1 to 2 μm long, and a human hair is very roughly 50 to 100 μm thick. Filter ratings, coating thickness and fiber diameters are usually given in μm.
Do not mix it up with the micrometer tool, the screw gauge used to measure thickness. Same spelling, but the tool is stressed on the second syllable (my-KROM-ih-ter).
Micron, micromillimeter and other names
Micron is an older name for the same unit and is still common in industry, as in a 5-micron filter. It used to have its own symbol, a lone μ, but the name was officially dropped from the SI in 1967, so write μm.
People also search for a micromillimeter. That is not a standard unit: you most likely want the micrometer μm, which is one thousandth of a millimeter, or the millimeter mm itself.
Watch the prefixes: mm (millimeter) is 1,000 times larger than μm, and nm (nanometer) is 1,000 times smaller. If μ cannot be typed, "um" is a common fallback in plain-text data, but avoid it in finished documents.
How to type μm
Type the micro sign and then m. On a Mac, Option + M gives µ. On Windows, Alt + 0181 (or Alt + 230) gives µ. In Word, typing 03BC and pressing Alt + X gives the Greek μ. In HTML, write µm or μm, and in LaTeX \si{\micro\metre} with the siunitx package.
Option + M and the Alt codes produce the micro sign µ (U+00B5), while a Greek keyboard produces mu μ (U+03BC). They look the same in most fonts and both are widely accepted; the micro symbol page explains the difference. The ㎛ character is a one-glyph version from East Asian fonts and is best kept for CJK text.