For instance, if I have a 24VDC/10A relay would running significantly less current (say 12VDC/2A) through it cause it to not operate properly?
For the actual contacts, it's a good idea to provide a minimum wetting current to maintain good contact (the datasheet will sometimes specify a minimum load current). Other than that you can run them at whatever current you like under the maximum value (10A in your example)
The coil should be driven at it's specified rating (24V in the example you link to)
You should really make sure that your relay has a decent datasheet that provides enough info - the one you link does not appear to have one, so I'd avoid it (unless you can google the part number and get it elsewhere)
Here are examples of the contact and coil ratings from a typical datasheet (in the contact section, note the minimum switching load and the different maximum switching ratings for AC/DC):
Coil Ratings (note the release voltage is much less than the operate voltage - once the relay has switched, it takes less power to keep it in position):
Varies by manufacturer and device. You would need to look at the datasheet for the relay to see if it is within range. By drastically underpowering a relay, you risk not triggering the internal magnetic coil.
Normally though, a relay stated as 24v/10A, would mean it is triggered by 24v on it's coil pins, and rated to carry 10A on it's nc/no pins. The 10A would not be it's relay trigger current, though there are industrial relays that can take that much.
If you are referring to the ratings for for the load (switching) path in the relay, you may run under the maximum current and voltage ratings with no problem.
If you are referring to the coil specifications, you must provide enough voltage to actuate the coil; this is often refered to as the "pickup voltage" specification.