Hard for me to recommend things on price as I'm in the UK but I'd agree with pretty much everything Jackrabbitslims says except the XT bit. I run Deore still and while it's heavier and uglier it works very well, the new M615 brakes are fantastic!
So hardtail, 27.5" or 26" (harder to find these days), Deore groupset if in budget and an air fork around 100-140mm travel. I think you'll struggle to get anything above Deore. 100mm would be fine for woods paths and general trail riding but less so on the steeper stuff, 140mm would be great for the steeper stuff but less good on flat path runs.
The frame geometry makes a difference as well. A slacker head angle will make going downhill easier but be more of a drag going uphill, the bike will be more likely to wheelie. What you will find is that most production bikes are geared towards the travel of the fork. So a bike with 100-120mm travel will have a fairly steep head angle whereas a bike with 140-160+mm travel will have a slacker head angle, so they are usually fairly easy to assess even if you don't have the measurements in front of you.
They'll be split into XC, Trail, All-mountain and Downhill or an amalgamation of the two. I ride a steel hardtail with a 130-160mm travel adjust fork. I run it at 130 nearly all the time on trail centre/off-piste enduro and then 160 when it's more like downhill (and by downhill I mean trails where many people ride downhill bikes with 200+mm travel at both ends).
I'd say in your budget a reasonable spec hardtail with 120mm air fork should be achievable, I think!
This is about right on budget (conversion from £ to $) and a very nice bike:
http://www.saracen.co.uk/bikes/trail/mantra-trailI'd actually be more than happy to ride that bike for the rest of my MTB riding days if it was all I could have.