Without spewing a whole bunch of jargon, most of which I don't have a clue about, at you, my understanding of impedance is that it's just a numerical rating in ohms about how effective a headphone is at producing db levels from any source. If the source is low-powered, like a personal discman or an iPod, than higher impedance cans like the HD580 will not be getting enough power to its high 300ohms and will therefore not be driven very loudly or as clearly --high and low frequencies will be rolled-off and recessed if the source isn't powerful enough to drive the headphones, thus the need for a dedicated headphone amp.
I've never heard the HD555, but from reviews that I've read, it's considered a very good headphone, especially for its price and is much easier to drive than the HD580. The HD580 is pretty much Sennheiser's top-of-the-line model and is universally regarded as one of the best headphones for under $200. The HD600 and HD650 are modified and enhanced versions of the same concept.