I'm looking into getting an oscilloscope, but I have no idea what the specifications mean (what they determine)... Can anyone suggest certain specs that I should look for when choosing an oscilloscope?
-Like, on ebay, most oscilloscopes that I found say 50MHZ, is that good, bad, or what?
50MHz is the maximum frequency the scope can measure and display.
For an Arduino running at 16MHz, with most of the signals much slower than this, 50MHz is adequate.
Higher is generally better though.
Dual channel scopes are better IMHO, allowing you to display two signals simultaneously. Storage (the ability to freeze signals) is extremely useful.
Great, thanks 
I'll keep that in mind.
Actually the bandwidth figure given for an oscilloscope measures the response of the vertical amplifier. It means that signals from DC to 50 MHz. will be attenuated a maximum of 3dB. That is the usual definition of bandwidth. As a matter of practical reality you can view signals of higher frequency but with decreasing fidelity since the vertical amplifier cannot keep up with them in either magnitude or phase.
That is the case for an analog scope. It gets really murky when you start talking about a DSO(Digital Sampling Oscilloscope). Here there be Tygers!
Oscilloscope Tutorial Part 1/3 - What is an oscilloscope? - YouTube he has 2 more vids about it