The isospin is a quantum number that distinguishes between different states of the same hadron. The concept was introduced to express why some hadrons are affected equally strongly by the strong force but had different electric charge. This was especially important for the proton and the neutron, which are nearly identical in all but a very small difference in mass when you exclude electromagnetism. In this context, the proton and neutron were considered to be the same particle, but in two different states. The state was expressed by the value of the isospin.
The "spin" part of the name comes from how Spin is able to couple two different particles together. In fact, isospin was intended as the explanation of why two particles seemed so bound to each other. Just like how a two particles of spin 1/2 could join to a system of total spin 0 or 1, the isospin would behave the same. The name is just a reference to the coupling behavior: it is not a type of spin or angular momentum in general.
Isospin is commonly reported as the total isospin and its -axis projection . The projection is often the important part, as that's the one that distinguishes between particles. The proton and neutron have same total isospin 1/2, but inverted isospin of 1/2 and -1/2. It is that distinguishes proton and neutron. A pair of the two makes a system of isospin 0 or 1.