That is a sometimes useful in itself, but almost always, op-amps are used with feedback loops. This means that there are external components, usually resistors and sometimes capacitors, to control the way it works. You can get much more useful effects from an op-amp by using feedback loops.
The feedback loops are really the key to using an op-amp. Over a wide range, you get very predictable operation from almost any op-amp in the same feedback loop. The differences start to show up at high frequencies, high accuracy, and high power conditions. That's where you look for specialty op-amps. Otherwise, any typical op-amp usually works for a standard circuit.
There are two meanings to this question. One is... how many kinds of op-amp circuits are there. The answer is, there are an infinite number, because you make those circuits by building feedback loops around the op-amp to perform various functions. Some examples of the most basic kinds are: inverting amplifier, non-inverting amplifier, summing amplifier, schmitt trigger, comparator, differential amplifier, integrator, etc.
The other meaning of this question is, how many kinds of op-amp components as a building block are there.
Well, there are many kinds of op-amp available commercially, made by companies like National Semiconductor, Texas Instruments, Linear Technology, Analog Devices, etc.
Some of these excel at high-speed operation, some excel at accurate low-voltage operation, some excel at driving power loads with their outputs.
Some are low-power. Some use FET inputs. Some use JFET inputs.
You can even build your own op-amp from a handful of transistors and resistors, or even tubes and resistors.
Go to Digikey.com or Mouser.com and search on op-amps, and see what you come up with. Thousands of different components.