What are the differences between the network peer levels? I've seen peer1 2 and 3 tossed around the web and wondered what the differences were.
I think you mean tier 1, 2, and 3 instead of peer. Peer1 is a company that sells transit service. Tiers are rough "levels" of internet providers that are largely self-determined by how close the company feels they are to the "core" default-free internet. Sprint, Global Crossing, Level3, and some others are the traditional Tier 1 providers. Anyone that buys service from those providers is a Tier 2, and so on and so forth.
You can look at http://en.wikipedia.org/wiki/Tier_1_network for a "contentious" view of what exactly constitutes a Tier 1 network, since a lot of this information is not publicly viewable. For example, a compny may buy a lot of transit leased line capacity (undersea fiber, for instance) and in exchange for that get free peering with that provider's internet network. In this case, they are not "paying for transit" specifically, so under some people's definitions, this might be a Tier 1.