This is a loaded question that requires a bit of a break down. It completely depends on the scheme, and in some cases I don't think that it is as clear as some of the answers given. IMO, the most important thing for a defense is probably the most basic: tackling.
Tackling is something that is being skipped over here, perhaps because of its simplicity. Maybe you all are looking at tackling as a given, something that you are assuming is already present in the defense. The reason I will claim it to be the most important thing, is because the best running backs in the league make defenders miss. If you can tackle surely, your defense can build around that.
The next thing you must look at is scheme; the type of defense. I find that once you have a scheme in place, the next most important thing is also simple, however at this point it differs at positions. Lets start with a 3-4 defense, and I can agree that the most important position on the field, can be found up front in any scheme. If you are running a 3-4, strength is key. You need a nose tackle that can battle with double teams, and win 80% of the time. This will require a lot of strength and stamina. In a 4-3 defense, you need players who are perhaps not as strong, but who are quick. They need to be strong enough to fight off double teams (although they will likely not see as much as a nose tackle in a 3-4), but that can get to the QB as quickly as possible, rather than relying on blitzing linebackers.
In a 3-4 defense, you'll need at least 1 linebacker that could double as a DE in a 4-3 defense. You'll likely be blitzing this guy all year long. Obviously this defense allows for a variety of looks as far as blitzes, using all 4 linebackers. In a 4-3 defense, you want linebackers that can simply cover their gaps. It becomes most important that they stick to their gaps, and clog the hell out of them. There will likely not be as much blitzing with 4 down linemen, so you need linebackers that are fast and sure tacklers.
DBs need to also be great tacklers. They will likely be the savers when it comes to this, seeing as they are the secondary. As far as pass coverage, I like DBs that don't loose sight of their man. More than defending passes and interceptions, I like a DB that makes the QB hold on to the ball, while the DL can get in the QBs face. So, pass coverage, being also simple, is the most important thing next to open field tackles for DBs.