Isospin reflects the fact that QCD says that the force between quarks doesn’t depend on what those quarks are - it’s the same whether they’re up quarks, down quarks whatever. Thus, from the point of view of the strong force, a proton (uud) and a neutron (udd) are pretty much the same. It’s therefore sometimes useful to think of them as merely two states of the same thing.
At this stage, you can introduce an analogy with spin. In QM, a spin 1/2 particle, when observed, will be in one of two states, spin up or spin down. Similarly, the isospin of a nucleon is 1/2 and the proton corresponds to “isospin up” and the neutron to “isospin down”. You can also extend the same idea to other sets of particles. Pions come in three types, with charges of +1, 0 and -1. Pions therefore have isospin 1, since spin-1 particles can have three states. All the maths involved is then identical to that for spin; nasty enough, but familiar to particle physicists. (Actually, there are pretty deep reasons why the maths here have to be the same, to do with symmetry groups.) If you want, you can think of an abstract isospin space in which a proton can become a neutron simply by being rotated.
The real usefulness of isospin is that it’s a conserved quantity in strong interactions and so is a very powerful tool for deciding which reactions are possible and the like. As you can probably guess from the fact that protons and neutrons are different, isospin isn’t conserved in electromagnetic interactions. Of course, historically isospin was introduced (by Heisenberg in 1932) long before it could be explained in terms of QCD; it was a handy approximation in understanding nuclear physics.
Hypercharge is just Y = B + S (well, it gets a little more complicated when you introduce charm etc.), where B is the baryon number and S is the strangeness. There’s nothing terribly profound: the textbook I checked this definition in just now (Halzen and Martin) even states “This choice has no physical significance; it simply centers the multiplets on the origin.”