Feel of the neck, check fit and finish. Look for string spacing and straight fretboard. Strum a couple chords and feel the headstock to see how it vibrates and sustains. I usually know what feels right in my hands. Couple of minutes I can say a maybe yes or definitely no.
Like a new pair of shoes, I kinda know in the first few seconds if there's a feel fit. I feel the neck profile/radius top to bottom, look at the quality & feel of the fretwork and the general quality of the build/assembly/finish, and play a few chords/notes up & down the neck to check the relative livelyness of the wood. I don't care for a guitar that's too plinky or too alive...these go back to the rack. Then, if it passes the previous tests, I plug in for the voice audition.
You can just feel in your hands when something is right or not.. and without the amp really isolates the guitar itself and doesn't let the loudness or feel from the amp cloud anything. The neck is the main thing I guess, like others have said. Gas-man sort of summed it up though, it doesn't take long. Unless checking for the minor things like dead spots or weird resonances on certain frets, otherwise it's a pretty quick thing.
To CubanB's point, unplugged you are able to truly evaluate just the guitar, free from the influence of an amp. I spend more time with it that way than plugged. You can pretty quickly determine if it has the resonance you want or if it's dead/flat sounding. I can pay more attention to the feel when I'm not worried about fooling with an amp. In fact I bought a CS 65 Strat recently without plugging it in at all and it has become my main player.
One other thing I do is press the upper body horn to my ear and pluck the low E string, so I can hear the sound of the body. If it has a good clear piano tone it will be a good guitar for me. If it's muddy or tinny, it won't. Very few guitars pass this test for me. This works for basses too.
Just the neck, the fretwork, and how the guitar balances. There's no point in playing it unplugged, as I don't buy my electrics based on how they sound without an amp. Even a fully-hollow archtop can be acoustically dead and sound marvelous through an amp.
Well if the hipster kiddies are not banging away on the line6 amps, i'll see if it plays well in my hands and can hear it acoustically. I'll inspect the neck if the strings are aligned, if the neck is warped, if it can keep tune, the entire setup.