Zero-sum means that DKP are distributed based on the value of items that drop. All members in a zero-sum DKP start with a score of 0. When an item is given to a raid member, she has to pay for it in DKP (going negative if neccessary). These DKP spent are immediately distributed among all the raid members. Thus the amount of DKP, i.e. the sum of all members in the system, is always zero. The idea here is that inflation is impossible, since no points enter the system from the outside.
In its pure form, zero-sum has the disadvantage that learning new content is not rewarded. There can be no bonusses for anything - punctuality, killing bosses etc. To a degree, this can be solved by "taxing" regular raids by giving a share of the points to a "point bank", which can later hand out points for wipe nights.
Another issue is that new players enter the system at 0 points, which is right in the middle of the veteran players. It is possible that a new member thus gets preference over a more veteran member in certain situations. This may be seen as a positive or a negative aspect of zero-sum systems, dependong on the guild's philosophy about loot. One possible solution is to specifically give newcomers lower priority on drops during their first few raids, which in an established raiding guild will almost certainly result in newcomers being low enough (by winning drops that noone else wanted) to not pose a "problem" to established raiders once their probation period has ended.
A last criticism aims at the inflation problem. The real disadvantage of regular DKP inflation is that after some time, the top players have earned so many DKP that new players entering the system have no possibility at all to ever reach their level (unless the veteran quits raiding). This same problem can also occur in zero-sum, if there are many inactive members with a negative score. To a degree however, it can be fixed by removing them from the DKP system and giving an equal share of their negative score to the remaining players.