To understand why an MGRS coordinate looks the way it does, it helps to know how the earth is divided. MGRS builds on the UTM grid layout and adds a layer of 100 km squares to it.
Grid zones: zone + latitude band
The earth is first divided into 60 UTM zones of 6° longitude each, numbered 1 to 60 from 180° W eastward. In addition, latitude bands run 8° high, designated with letters from C (at 80° S) to X (at 84° N). The letters I and O are skipped to avoid confusion with the digits 1 and 0; band X is 12° high instead of 8°.
A zone number plus a band letter together form a Grid Zone Designator such as 31U: the large square within which the rest of the coordinate applies.
100,000-meter squares
Within each grid zone the area is further divided into squares of 100 × 100 km. Each square gets a code of two letters: the first gives the column (easting direction), the second the row (northing direction). That way, with just two letters, you know which 100 km square you are in, without mentioning large numbers.
Why letters instead of numbers?
By expressing the coarse part of the position in letters, the coordinate becomes shorter and less error-prone. The large UTM numbers (hundreds of thousands of meters) are "hidden" in the zone and square codes; you only write the fine position within the square as digits.
Boundaries and exceptions
Around Norway and Svalbard some zones are widened or narrowed, and the poles use a separate system (UPS instead of UTM). For most users those exceptions don't matter — a good app takes them into account automatically.
Seeing the grid layer in MilMap
MilMap shows the MGRS grid lines over the map, so you can see at a glance which grid zone and which 100 km square you are in. Handy for orienting yourself and for quickly checking coordinates.