I think I have a few things I look for, in this particular order:
English willow
Size - As long as the bat is the same length as a regular short handle
Profile & concaving - Middle cannot be too high. I look for anything from low to mid, occasionally mid-high, and no concaving is preferred. If there is any concaving, it has to be only very slight
Handle shape - Must be oval. I think this might just be from reading about oval handles. I'm sure that if I didn't read about them and the benefits, I reckon I would be perfectly happy to use a round handle. Also, the handle has to be relatively thick! I don't like handles too thin.
Weight - I used to look for bats around 2.7-2.10, with 2.10 being really heavy, although I now generally look at slightly heavier bats now, but will be happy for anything between 2.8 to 2.14, with 2.11-2.12 being the ideal weight. Funny thing is none of the bats I would consider being my match bat fall in this ideal weight range at the moment! The 2.14 I use actually feels alright for me to use, if anything, it feels just a tad heavy. Bat speed isn't quite up to what it used to be when I used a 2.8, but it actually limits my shots to the ones I'm good at, and I can actually pull the ball now. Cutting is not easy with a heavy bat for me though!
Looks - Too grainy can put me off, and too much heartwood can too, although I did just buy a Talisman with 18 grains and more than half heartwood

Generally, I look for all sapwood and anywhere between 7-12 grains, with 8-10 being the ideal range. Would happily buy bats with 4 grains if I really liked the profile!
Pricing - Has to be a good deal, or a fair pricing at least. Mind you, I used to get all my bats at RRP at my closest cricket retail store... An Adidas Incurza club for £200 when it came out...

Good bat, but not a good or fair pricing for it!
Stickers - Despite this being the last thing I look at, this can put me off if I'm honest!
Those are all the things I generally look for when I'm buying bats. If I'm going for bats crafted to your specs, then I'm slightly different, but generally along the same lines as this list! I can be a pain in the ass to work with!