Some thoughts
You've got lots of paths. Some of them have the same initial and terminal points. Some of those can be continously deformed into each other. Those are path homotopic. If they are path homotopic. For the purposes of homotopy theory, they are considered the same. Consider the punctured plane. That is, R^2 minus the origin. Now, any paths in the punctured plane that have the same initial and terminal points form a "loop" if you look at "the picture" If the origin (which is "missing" in the punctured plane) is inside this loop then the paths are not path homotopic, because you couldn't continously deform one of them into the other without dragging it over the missing origin. The fact that they're not homotopic tells you that there is a "hole" in the space, that its not simply connected.
Now there are a lot of possible paths that could be in any space and so you organize them. From the example above it should be clear that loops can tell you about the "holes" in a space. So why not study those paths that start and end at the same point, i.e. loops. The particular point where they start and end is called the basepoint. They're kind of like lassos. So you plaster your topological space with all these loops and you cinch them tight. Anything that you can "cinch" all the way back to your basepoint isn't very interesting. If you can "cinch" (and by "cinch" I mean continously deform) the loop back to the basepoint then that loop is homotopic to the constant loop which starts at the basepoint, goes nowhere, and finishes at the basepoint. But if your space is interesting, some loops will not be null homotopic (won't be cinched back to the basepoint). These will get "caught" or "hung up on" holes in the space. Of course there will be lots of different loops around the same point. More technically, there will be lots of path homotopic loops around the same point. Now there's no need to repeat information. If you found a hole, you found a hole, you don't need to find it over and over again. So you make equivalence classes based on path homotopy of loops.
Now because these are loops, and because they are all set at the same basepoint (so they all start and end at the same place) you can "add" them by doing one loop and then doing another. Just like Mathwonk said, you add them by doing each one twice as fast so that you make a new loop, one that starts at the basepoint, goes out around path number 1 comes back to the basepoint and then heads out again for a trip around path number 2, finally returning to the basepoint. So this loop looks like a pair of petals on a flower, but its a loop nontheless.
Now, as you'll know from reading whatever you're reading (I like Massey, personally) "adding" equivalence classes of the loops in this manner is well defined (if you don't know you should prove it). And you've got inverses (going around the loop in the other direction), an identity (the null loop which is the basepoint) and so you've got a group.
now the alpha you've been talking about is a group homomorphism as Matt Grime said. Look at what its doing. Its mapping one fundamental group to another fundamental group. Both groups are concerned with the same topological space X, but have different basepoints. What your book is saying is that it doesn't matter what basepoint you choose (as long as your space is path connected). This is good, it means we don't have to worry about missing information if we pick a particular point to base our loops at. This what it means for alpha to be an isomorphism. For a path connected space, the fundmental groups at each basepoint are all isomorphic.
Well, that ought to keep you busy for a while. Try to organize your facts and theorems and distinguish between notations. As the other contributors have mentioned, alpha is not a path, it is not an equivalence class, it is a function between groups that preserve the group structure, i.e. a homorphism.
cheers,
Kevin
