Tier is an adjective which (loosely) describes where in the hierarchy a network fits.
There are as many definitions of tiers as there are network operators, but very broadly, there are three major ones, for which I'll invent new (:-) names, so as to stay out of the fray:
These descriptions are really applicable as much to a specific router as to a complete network: many larger providers operate at more than one level simultaneously, and not all make any administrative distinction between them.
The backbone networks are those which are default-free; their routers do not have a default route, and must therefore have an explicit route to any network to which they must route packets. Backbone networks may peer with one another. Backbone networks are rarely connected to directly by end-user loops, though these customers may connect to end-node networks operated by the same providers; the borders are not always clear, administratively.
Mid-level networks are those which, for lack of a better description, sit between end-node networks and the backbones; the primary distinction is likely that they have default routes.
End-node networks are those to which end-user customers connect. Those who feel that it is incumbent upon "ISPs" to take actions (such as packet filtering) to prevent Tragedy of the Commons on the Internet usually are (or should be) talking about this category of network.