Measuring distances along a straight line is very easy: measure it on the map with a ruler and multiply it by the divider of the scaling factor: for instance, 12 cm on a 1:250'000 map is
0.12m x 250'000 = 30'000m = 30km.
You can also compare the measured distance on the graphical scale bar. If you don't have a ruler, you can use any object (a pencil, a small stick, your finger) to report this distance on the graphical scale bar.
Measuring distances along a complex path (a broken line) is a little bit more complicated: You can divide it in small linear segments, measure each segment individually on the map, then apply the multiplicating factor. You can also use a thin rope (or a long hair!) to measure the length of the path on the map, then apply it on the graphical scale bar.
A more comfortable (and accurate) solution is to use a curvimetre. This special device is actually based on a very small wheel that you move along the path on the map. It will indicate the length of the path, and as most odometers are used with map, they often give you a direct reading of the actual distance, for certain scales.
Some users will have to measure surfaces on a map. To do so, you have to divide the area to be measured in simple polygons (rectangles and triangles) and to measure the surface of each element on the map (in mm²). The surface measured on the map must be multiplied by the squared value of the divider of the scale factor to obtain the actual surface of the polygon. For instance, a 56mm² area on a 1:50'000 map corresponds to: 56 * 50000² = 140'000'000'000 mm² or 140'000 m², or 14 hectares.
(1 m² = 1'000'000 mm / 1 ha = 10'000 m² / 1km² = 100 ha)