There are actually two kinds of isospin:
The name isospin implies a relation to the (perhaps more familiar)
spin - that is only partly true. Isospin does not have anything to do with
angular momentum, however it has the same mathematical properties as spin. That is where the peculiar rules for
addition come from: Spin is a
vector. That means if we have two spins we want to add, the result depends on their relative orientation - if they point in the same direction, we get T1+T2, if they point in opposite directions we get |T1-T2|. Everything in between is also possible, but keep in mind the
quantization rules.
Now in
particle physics you say there is a thing called a
nucleon, which has a strong isospin with an
absolute value of 1/2. Which direction is it pointed? Well, that depends. You take an arbitrary direction (usually the z direction in our x-y-z
coordinate system) and look at the component of I along it. Because of quantization this
quantum number I
3 (which is a
scalar, as opposed to the vector I) can only be -1/2 (if I is pointing 'downwards') or +1/2 (if it's pointing 'upwards'). A nucleon with I
3=-1/2 is called a
neutron and a nucleon with I
3=+1/2 is called a
proton.
But a nucleon consists of
quarks! A proton is a combination of two up quarks and one down quark, denoted as uud, and a neutron is udd. It turns out that on a deeper level we can assign a strong isospin to
quarks as well - the u quark gets I
3=+1/2 and the d quark gets I
3=-1/2.
We can do the same for
leptons. They do not interact strongly, the weak force does affect them though. The
weak force is a bit strange, because it differentiates between left and right (see
CP violation, or rather
parity violation). So we say that the
left-handed electron and the left-handed
electron neutrino are a weak
isospin doublet, ie T=1/2, the electron has T
3=-1/2 and the neutrino T
3=+1/2. Left-handed by the way means that the spin vector is pointing opposite to the
momentum vector. There is no right-handed neutrino (which was indeed a puzzling discovery - but since it appears that neutrinos do have some mass after all, it is no longer strictly true, see
helicity conservation), and the right-handed electron gets assigned T=0.
So far so good :) Are you still with me? Now it gets even more confusing. In addition to their strong isospin, quarks have weak isospin too. Because the
eigenstates of the strong interaction and the weak interaction aren't quite the same (see
Cabibbo rotation and
CKM mixing) we have to "invent" a new quark, called d'. But the rest stays the same: u and d' are a weak isospin 1/2 doublet with T
3=+1/2 and T
3=-1/2 respectively.
So what's it good for, you ask? After all, it seems more than a little arbitrary! Well, it lets physicists formulate new
conservation laws and we can use those to make
predictions about
experiments, eg which particles may be created in a certain reaction. So far, the predictions of the isospin formalism have been right, and that's what justifies it.