There are so many different amplifiers because there are so many different applications, each requiring different attributes.
A perfect voltage amplifier, for example, has infinite input impedance and zero output impedance. Neither of these exist in a real amplifier, of course, but there are devices with Gigohms of input resistance (look for a JFET input stage device, for example).
A perfect differential amplifier (within which opamps exist) has zero offset between the inputs, but once more, no such device exists. This Vos term is always stated in datasheets. In a high gain situation this input offset appears as an output offset voltage given by Vos * gain. Where tiny signals must be amplified (as in a strain gauge for instance - there are numerous applications here) we would use what is known as an Instrumentation Amplifier which is a device optimised for high gain, low offsets and high CMRR amongst other things.
Perhaps you are in a battery operated system and need a low quiescent power amplifier. The trade-off here is usually Gain Bandwidth Product although advances are still being made in this area (as it is in all areas, but with the Internet of things appearing, this is becoming a key driver).
Perhaps you need a really fast Video Amplifier for clean amplification of video signals.
There are High Speed amplifiers, optimised for GBW. Then there are Zero Drift amplifiers.
By now, you may be getting the idea that so many different types of device exist for the very good reason that each type has been optimised for a particular task: which one I choose depends on the specific requirements of the application.
I have barely scratched the surface here, but I suggest looking through the tables for these devices at Linear Technology, Analog Devices and Texas Instruments for starters. Maxim Integrated also has an excellent series of tutorials ans application notes.
All these manufacturers have excellent tutorials and application guides that are a wonderful resource for anyone wishing to learn about these devices.
As noted by Adam Haun, there are amplifiers that are designed to have a minimum gain well above unity; the advantage here is better transient response as the Dominant Pole (scroll down in page) can be at a higher frequency, maintaining more bandwidth at lower frequencies. These devices are not unity-gain stable, and therefore may not be used at lower gains.
Edit. Added current feedback amp: Thanks for the reminder, LvW
A typical current feedback amplifier has incredible slew rate. This one is listed at 1600V/us which while available in many devices, is still truly astounding in an 'ordinary' amplifier. Although these devices can be a little more difficult to understand at first, they have significant advantages when used appropriately. Read this application note for a good example.
Another update
As can be seen from the comments, amplifiers come in many different flavours, such as differing output drive current, rail to rail outputs (many can only go to within 1.5V of the power rails), rail to rail inputs (usually requires a dual input stage which can have its own very peculiar effects), yet others optimised for high side sensing (the common mode range can exceed the power rail) to name but some possibilities.
There are others designed for ADC interfacing that can have an external common mode reference so that the output is always centred in the middle of the ADC range, truly fully differential devices and more.
The range of amplifiers available is truly enormous. What was a 'general purpose' amplifier 20 years ago is now 'low end' although as noted many older amplifiers are still available, for a number of reasons.
The key to choosing an amplifier is deciding which features of the perfect op-amp you need to most closely get.
One way to become familiar with amplifiers without actually making the circuit (at first) is to use a simulator. There are a number of these around, such as LTSpice and there is a free version of Simetrix (node limited) amongst others.
When using simulators, you will need to understand the limitations of the models used. There is an excellent application note at Linear Technology that shows what the models really contain (and it is not the actual circuit in the amplifier).
HTH
Best Answer
Before be an engineer, I was also an electronics technician. In a similar way to the course that you teach, the objective of my one was to enable technicians for the industrial segment. Our discipline was about linear and non-linear applications of Op. Amps. For the former case, for example, are the voltage follower, current sources, current amplifiers, voltage regulators and DA converter. A good topic to address are the single supply power configurations (avoiding CMR violation). For the later case (important for the industrial area) are the Schimitt-Trigger, waveform generators - square and triangular, window comparators, etc. A good topic to address is the internal (simplified) model of 555 IC. In the mid are the precision rectifiers.
EDIT
As a practical example of waveform generation with "some" additional glue, consider the high level circuit shown below. This is a way to generate a simple lighting pattern for a water fountain. The figure shows three LEDs (R, G, and B), but my original design drove three 12V high current water-proof incandescent bulbs (one 127Vac/12Vac transformer, three TCA785 ICs and three TRIACs completed the circuit). The complementary E1 and E2 (staircase) inputs are produced by means of 555 oscillator, one R2R network (D/A converter) and two LM324 op. amps. The switches S1 to S6 are implemented with 4066 analog switches. A set of flip-flops and logic gates (mealy machine) generates the signals for 4066's. The challenge was not to use microcontrollers twenty years ago.