This is the one thing that stuck in my mind after the two back-to-back classes yesterday. Essentially, it's the set of circles centered at (1/n,0) with radius 1/n for all n in the natural numbers (the counting numbers, n=1,2,3,...). It's actually a quite beautiful thing. Take a peak! (Thank you wikipedia!)
Now, the one thing that got me during class was that my professor contradicted himself. He first made the point that this is very similar to the wedge of circles (but that it is not homeomorphic - their topologies are very different). The difference in these two structures? Well, visually, we can see that the Hawaiian Earring are circles within circles, adjoined at one vertex. The wedge of circles (commonly referred to as the rose) are a ring of circles adjoined at one vertex. So, there's that difference. It doesn't seem that the rose contains strictly the circles of the same radius, so the size of the circles does not seem to matter.
Then it got confusing. A point was made that the Hawaiian Earring (X) is compact, and that it is not a CW-complex. But, then he went on to say that one vertex and infinitely many edges implies that the space is not compact. Yet, X has infinitely many edges and one vertex, so why is it compact?
After some research, it turns out that X is compact. And, I'm pretty certain that the rose is not compact, although I still haven't found any literature on it - I'll be hitting the library later on.