Usually it's summarized as "internet providers want to charge different companies different rates", as if that's self-evidently a bad thing. But you could imagine a VOIP company reasonably willing to pay extra to get different QoS guarantees, or a bulk downloading service accepting a higher-latency cheaper service. After all, companies already must pay for their bandwidth, and I'm sure the contracts they negotiate aren't all the same. Also consider airlines, which charge varying amounts for effectively identical service in an attempt to discriminate between customer types. Both of these are the sort of thing that is typically solved by markets; for example, a highway system with varying tolls will cause different users to plan different routes depending on their needs and willingness to spend. As far as I know, this highway analogy is exactly how backbone internet routing works -- different companies pay different amounts to peer with different providers.
So here's some ideas about what I really think is at issue (which, unfortunately, can't be summarized in a nice buzz clip by Jon Stewart): the market for "last mile" broadband is not competitive. I can't find it now, but I recently saw some sad numbers on which fractions of Americans have access to different broadband providers. You've probably experienced it yourself, in any case -- most people have only one or two choices. If for this last leg of the trip the tolls are prohibitive, especially when the businesses who determine the costs are also providing competing internet products, the issue becomes much more troublesome. More broadly, the fear is that ISPs will use varying price as a weapon to stifle competition and consequently innovation.
In that light this debate becomes like many others we're currently faced with, where there's a tension between the incumbent's power to repress change and the potentially valuable upstarts. For example, this is similar to the tension in patents: in a working patent system (which ours is not), there's a reasonable argument to be made (in fact, the argument that is made for why patents should exist) in that granting a limited monopoly motivates the original investment of work. I think this is where that quote that set all this off came from: "Now what they would like to do is use my pipes free, but I ain't going to let them do that because we have spent this capital and we have to have a return on it."
But much like the patent system, part of government's role should be to ensure efficiency (if you're not convinced that the patent system is a wreck, see Bill Gates' infamous quote). Going back to highways: if someone could monopolize both the 101 and the 280, Silicon Valley would come to a standstill, so we instead agree to let the government manage the freeways and we all share the costs. Lessig makes another analogy that makes what's at stake more apparent:
Imagine there are only two television stations in a particular democracy. They both begin to "access tier" — charging different rates to different political candidates. So Dems get a rate 1/2 the rate charged to the GOP; or major parties get a rate that is 1/3 the rate charged to Independents. Does that begin to trouble you?So the question underlying the debate is whether the internet is a public service ("commons") to the point where the government needs to step in and subject it to common carrier laws. At that point this appears more like a debate, in that reasonable people can disagree -- after all, people still argue we ought to privatize the highway system. But I can now also see why so many groups whose opinions I generally agree with (Lessig, Felten, Google) all agree on this issue as well.
1 See also.