The two most important specs are forward current and reverse voltage. I usually buy a higher voltage version (1N4007) since they don't cost much more and I'm usually buying more than I need so I don't know what I'll be using them for.
As a flyback (or flywheel) application the (peak) current through the diode will be the same as the current through the motor/coil, and the reverse voltage will be the applied voltage.* (So, in most low-voltage applications, voltage isn't an issue and you just have to check the current.)
I will be using it sometimes for rectification (max 24V)
When I'm building a power supply I usually use a [u]bridge rectifier[/u] which is 4 diodes in one package. I may have made a bridge from 4 1N4001's, but I don't remember. With a single diode, the power is "off" for half of the AC cycle which means you get more ripple (or you have to use a bigger capacitor) and you can only get half the current.
Or, if you have a center-tapped transformer you can get full-wave rectification with two diodes.
The forward voltage across a standard silicon diode is about 0.7V (depending on current) and in some applications where you want a smaller voltage drop, you can use a Schottky diode. Schottky diodes also switch faster than standard silicon diodes so they are sometimes used in high-frequency circuits.
- When you disconnect an inductor/coil you get a high-voltage kick (back EMF). But, since the diode is "backwards", that back EMF becomes the forward voltage across the diode. That "kills" the high-voltage kickback, and your diode never sees high voltage.