I have a device that has an analog output that I would like to measure it with a 5v Arduino. The output is rated 5-50mA.

How would one go about measuring current? A resistor across the terminals and measure voltage across the resistor? Am I on the right track?

 

Yes, thats the way it's usually done. A 100 ohm resistor will have 5 Volts across it at 50 ma.

 

That's easy enough.

Now I'm just curious why would some devices have voltage outs and some amps outs? What are the ups and downs of using one over the other?

 

With current loops (current output) the accuracy of the output is not effected by the voltage drop in the wire..
https://en.wikipedia.org/wiki/Current_loop

 

That's easy enough.

Now I'm just curious why would some devices have voltage outs and some amps outs? What are the ups and downs of using one over the other?

What type of "devices" are you talking about.

 

What type of "devices" are you talking about.

An HVAC environment monitor that might hang on a wall hundreds and hundreds of feet away from the main control panel.

 

For that a current-loop (commonly 4-20mA) is often used since it's less sensitive to noise pickup then a voltage signal.
Also the same current can be used to provide power to the remote sensor, so two wires can provide both power and signal.