Relay and Transistor Questions

For a given relay, when it says (don't exceed 24V/5A), what is it referring to? The current/voltage across the Common / NC,NO connections? Or the input? (Example: https://www.sparkfun.com/products/12093)

Also, I thought I had a handle on transistors, but I don't. For a given transistor, how do I figure out how much current / voltage drop I can have between the collector and emitter?

Thanks!

Your first question is answered in the link you provide:

gives your Arduino the ability to control a single electromechanical relay for switching loads of up to 24VDC or 40VAC.

So, is "The current/voltage across the Common / NC,NO connections".

For your 2nd question (short version):
voltage drop I can have between the collector and emitter: 0V (is at the saturation state);
how much current: Ohms law (if you know the resistance of the relay coil an the voltage that you supply to it)

Look at Vce: Saturation for the NPN (typically) transistor you want to use.
It wll be in the 0.3V to 0.7V range.

If it's an N-channel MOSFET, it will have an Rds rating with Vgs = 4.5v. You want that nice & low, like < 1 ohm. Good ones will be well under, say 100mOhm (0.1 Ohm)
Then if your coil needs say 50mA to turn on, the voltage across the Source:Drain will be
50mA * Rds (V=IR, Ohms Law), so .05 * 0.1 = .005V.
And the rest of the voltage will be across the coil.

osmosis311:
For a given relay, when it says (don't exceed 24V/5A), what is it referring to? The current/voltage across the Common / NC,NO connections? Or the input? (Example: Relay Shield - DEV-12093 - SparkFun Electronics)

Those specifications are the limits of the relay's contacts only (C//NO/NC).

Also, I thought I had a handle on transistors, but I don't. For a given transistor, how do I figure out how much current / voltage drop I can have between the collector and emitter?
The datasheet for the transistor will state the maximum collector voltage and the maximum emitter/collector current. However one never runs a transistor at it's maximum rating. So for example a 5 volt relay coil might rightly use a transistor rated at 40vdc or more maximum collector voltage and 1 amp or more E/C current even though coil may only draw 100 mA.
Thanks!