Are electrodes mostly homogeneous or are there variations by usage, such as EEG vs EKG/ECG?
Electrodes can be different in shape, depending on their purpose. For example, EEG electrodes tend to be cup-shaped. The cup can then hold some conductive paste, improving signal-to-noise ratio. ECG electrodes on the other hand tend to be patches that can be used on larger surfaces, such as the chest. The conductive material is then mixed with the sticky layer. EMG electrodes tend to be similar to ECG electrodes, if not the same.

That's for "wet" electrodes. Now on to dry electrodes. These use different ways to conduct signals. Some require you to apply a conductive paste for increased signal-to-noise ratio. Others have a pin-like structure that pokes into the upper skin layer. They can come in a variety of shapes and sizes, all meant for improving signal-to-noise ratio (and all of course claiming they're the best). And again others are simple metal surfaces that can be attached to conductive materials of your own choosing. The latter can be found in the increasingly popular ECG chest straps for athletes.

And last but not least, there's the active electrode. These electrodes have tiny circuits in them that amplify the signal and filter out common noise. These can be seen below. Perhaps the "special" electrodes are active ones? 



As far as materials go, there's not much difference really. Some are slightly better than others at transporting signals, but unless you're ordering from China, you can rest assured most electrodes will conduct the signals they are intended to conduct.

Real question is, what are you wanting to use your electrodes for?