Tackling: As a defensive midfielder, the defence needs to be helped out. Against swift attacking midfielders, the man needs to have good Strength to take the ball away. Tackling is a vital skill
Passing: The link between defence and midfield, perhaps even attack (in the case of players like Pirlo and Carrick), a wide range of passing is key. Good vision and execution from a defensive position is very hard to defend against for the opposition. A bit of Creativity wouldn't go amiss.
Marking: Marking that surging run from midfield can be crucial in prevent an opposition goal. If a defensive midfielder is set to tightly mark someone like Kaka for instance, it can work to great effect, rendering the attacking player almost useless. Positioning is a good skill to have here.
Heading: A long kick upfield by either goalkeeper is there to be won in the air. Since the midfielders are of course, in the middle of the field, it makes them ideal candidates to get onto the aerial ball. It can then be flicked to a more creative player who can dictate the play. Jumping is a big bonus to have for Heading.
Long Shots: In this position, you don't usually venture forward as much as the person you're holding for (the attacking midfielder). However a late arrival into the area often happens, and the ball comes across to your feet. Having that long shot in your drawer can give your team that winning goal. Michael Essien has done it on occasion.
Technique/First Touch: Some neat play from the back, leading into the middle of the field, and maybe even up front is good for a team. It's not completely necessary of course.
Determination/Work Rate: Midfielders get up and down more than anyone else perhaps. For this you need a drive to go forward and backwards again and again and again. Determination needed for a crunching tackle is just as important as determination to smash the ball into the back of the net. Flamini is a good example of someone who covers a hell of a lot of ground in a match. Stamina is also needed of course.
Bravery/Aggression: A strong, physical presence protecting the back four is not a sight attacking players like to see. Worse when they're coming straight at you. Good tackling sometimes depends on it and is a stat that's most important in defensive midfield. Having good Tackling is a bonus.
Decisions: The middle of the park is the most common place to lose the ball to the opposition. It often happens because of a mistake, a poor decision. Decision making is key to a midfielder's game.
Teamwork/Influence: This position requires the player to demand the ball. A strong presence. Someone who pulls the strings (in some cases).
Positioning/Anticipation: Although it's far more important for defenders to have a good positioning sense, it's good to have someone who can help out, puts pressure off defenders and intercepts the ball further up the field in a less dangerous position for the team.
Creativity: A player like Pirlo is a perfect example of a creative "defensive" midfielder. Although lacking in several defensive stats, he is a deep lying playmaker, pulling the strings, providing the passes from a position where few opposition players can get near him.
Stamina/Natural Fitness: A requiremen of this position is to get up and down and up and down every minute of the game. A defensive midfielder can't get tired; he needs to be the team dynamo. Determination and Work Rate are related to this as well.
Strength: Winning headers, winning tackles, muscling off the opposition - all parts of a DM's game.
Jumping/Balance: Winning headers and keeping on your feet benefit the team. If the attacking player gets behind/around you then it puts the defence under pressure. Being solid is what's required.