Grid systems vary, but the most common is a square grid originating at the bottom left of the map. Each grid line is numbered sequentially. The grid numbers on the east-west or horizontal axis are called Eastings, and the grid numbers on the north-south or vertical axis are called Northings.
Grids may be arbitrary, or based on specific distances -- for example the 1:50000 Ordnance Survey maps use a 1km square grid spacing.
A grid reference locates a unique point on the map. The resolution of the reference provides for some given margin of error -- for example a simple town plan may only use a simple grid system such as letters for the Eastings and a single number for the Northings. A grid reference in this system, such as 'H3', only locates a particular square rather than a single point, but you will usually spot what you are looking for, so absolute accuracy is not required. For Ordnance Survey maps, a six-figure grid reference is usually used, providing a resolution down to 100 metres on the ground.
To give a grid reference, quote the Eastings first, then the Northings. For example, a reference of 696018 specifies square (69, 01), and the point (6, 8) within it. For casual use, estimating the tenths by eye is usually sufficient, but for complete accuracy, a romer is used to mark the point exactly.
For example, the grid reference above, 696018, is the position of the church at Little Plumpton, in the fictional illustration below.