Mohs Hardness Scale is basically a measurement of the hardness of a material, or more specific, a rock or mineral. But I have never been able to figure out why some minerals are harder than others.
The hardness of minerals is diagnostic because the hardness is determined by the strength of bonds and the structure of the mineral lattice. Hardness is basically the stress required to create and grow extended lattice defects such as micro-fractures, stress twins, and dislocations.
Diamond, quartz, and framework silicates, such as feldspar, are hard because they have a three-dimensional lattice of covalently bonded atoms. Native metals are soft because they have close-packing lattices of metallic bonded atoms.
Hardness is an intrinsic property of a mineral like density or refractive index.
There are some subtleties that I'd like to add, in addition to Mark's answer. When talking about the hardness of a mineral, the nature of the chemical bonds in the crystal structure (e.g. covalent vs ionic) are not the only important thing. Crystal morphology is also important.
For example, Si-O-Si and Al-O-Al bonds usually cause minerals to be hard, such as in the case of quartz and feldspar. However, if you look at clays and other related minerals, they also have similar bonds. In this case, however, crystal morphology is the determining factor. Clays are not one big crystal, but an aggregate of many micron-sized crystals. The hardness is not determined by formation of crystal lattice defects and dislocations as in Mark's explanation, but rather by movement of individual crystals. Assuming you could have had a single huge clay crystal, it would be much harder than the common lump of clay.
Also, take for example graphite. One of the reasons for its softness is the fact that each carbon ring (see link for refresher) slides on top of another, and not any change in the actual bonds of the carbon atoms.
Calcite is another example. With a Mohs hardness of 3, you are not supposed to be able to scratch it with your fingernail. However, chalk is a calcite bearing rock sometimes defined as a "limestone you can scratch with your fingernails". The case here is similar, individual clay-sized grains of calcite are moving around.
Another point is that the Mohs' scale is sometimes too idealistic. Take for example feldspar, that has a Mohs hardness of 6. This is for a perfect crystal. However, often feldspar crystals are slightly altered to said clays, thus reducing the hardness.
An additional interesting point is that per crystal, hardness is not always a constant. Take for example the mineral kyanite. It has a hardness of 4 in one direction and 6 to 7 in another direction.