I look at [non]conservative forces in terms of what they turn energy into. When moving against a force, you do work; when moving with a force, work is done on the object. In a conservative forces, these are both "efficient" energy conversions: 100% of the work you do is turned into usable potential energy, and then 100% of that is turned back into kinetic energy of the object. This is the case with, say, an object falling under gravity in a vacuum: its energy is exactly the same at the bottom as the top.
Friction does not do this. Friction, by definition, converts all of the work done into heat. The force opposing movement in a certain direction $x$ is the sum of the work required to get the potential energy, and the energy that will be lost. In friction there is no potential energy stored, so you are always doing work to move.
The force of your arm on an object is also not conservative. Perhaps, with very efficient muscles, you can lift a weight 100% efficiently. But when you lower it again, the potential energy is not converted back into energy in your arm. In fact, you'll have to start firing your muscle cells to slow it down as it falls, which will waste heat, so you do work even when it's coming down. Again, all the work is turned into heat.
As an example of something in between, an object flying through air under gravity: going up, most of its energy is turned into potential energy, and a bit into turbulence in the air (and eventually heat); and going down, most is turned back into kinetic energy. So perhaps, upon reaching its original height, it has 90% of its original speed. This force was mostly conservative.
Why are there "2 kinds of forces"? In the same sense that there "are two kinds of English sentences": true and false. Conservatism is just a property I can talk about, indicating 0 loss. This is the same as "[perfectly] elastic" and "inelastic".
Why are the 4 fundamental forces conservative? -- because they have nowhere else to put the energy! On a microscale, all energy is "tracked", including heat, as a the kinetic energy of particles creating heat. By definition, energy is the thing that is conserved by all the forces. So we built energy such that the fundamental forces would be conservative, and this determined it. Then, if you choose to ignore certain kinds of energy (such as heat), you can get nonconservative forces.