Universal (magic) Op-Amp Fundamentals
simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. The universal op-amp and variations &quot;with bits left out&quot;.

What is Ag, Rg, or Ai?

This has been addressed in some of the other answers.

Also, what is the purpose of these types of Op-Amps?

There's nothing special about the op-amps. It's the way they're used that's special - although there's nothing special about these configurations these days.

They look very similar to summing Op-Amps.

The op-amp is the part signified by the triangular symbol. The layout of the inputs and feedback make it into a certain type of amplifier.

Also where it says, &quot;This formula subsumes all previous configurations,&quot; the professor had just gone over five standard Op-Amp Configurations prior to this one.

And now s/he's telling you that this one configuration includes all the others. By tying certain inputs to ground or leaving them floating you can create any of the other five configurations.
Texas Instruments make a Universal Operational Amplifier Evaluation Module which allows easy creation of all of these configurations as well as some filter circuits while using surface-mount versions of their op-amps.