The greater dispel magic spell can dispel one specific spell on an area, creature, or object per 4 caster levels its caster possesses. As per the dispel magic spell, the caster must name the specific spells that will be targeted in this way, presumably not being able to pick the same spell twice and presumably before rolling the check. Then, in order from the highest spell level to the lowest, the check is compared to each spell to see if its dispelled or not.
(As I was reminded in answer to this question, Pathfinder vastly simplified from its D&D 3.5 antecedent here the rules for a targeted dispel magic spell. In Pathfinder when a targeted dispel magic spell is cast, the caster makes but one dispel check.)
Example
Kram's combat efficacy is impaired after having been affected by the spell grease (a 1st-level spell), the blindness effect of the blindness/deafness spell and the deafness effect of a separate blindness/deafness spell (both are 2nd-level spells), the paralysis of a hold person spell (a 3rd-level spell), and the forced friendship of a charm monster spell (a 4th-level spell). Ekaj the Sor10 wants to cast a greater dispel magic spell on some of the spells affecting Kram.
Under what I suspect is the typical reading of the greater dispel magic spell, the caster—before making the check—names each spell the greater dispel magic spell will attempt to dispel. I also suspect that the typical reading permits picking each spell only once. Thus, from the example, Ekaj says, "I cast greater dispel magic on the following spells affecting Kram: charm monster and hold person." Ekaj makes the check; the GM compares the result in order from the highest spell level to lowest to the spells Ekaj picked and notes which spells are dispelled.
Why do I suspect that this is the typical reading? In my opinion, this process is more intuitive, linear, and accessible than the alternative. (You, dear reader, may have different opinions on what is intuitive, linear, and accessible; that's cool.) However, just because that's what I believe to be the typical reading, that doesn't mean it must be your reading.
Alternatives
According to the text the caster need only name the spell that he's attempt to dispel. The GM may agree that this means that the caster need only name the next spell he's attempting to dispel. Thus, for the example, Ekaj would say, "I cast greater dispel magic on the charm monster spell affecting Kram." Then he could say, "For the second spell, I pick hold person," but he could've picked any spell of a spell level lower than charm monster. The risk with this approach is that a GM's campaign may not have spell identification include caster identification, and some casters have spells at different spell levels. The GM must decide what happens if the caster makes a mistake and picks spells out of spell level order.
Because Pathfinder only has the caster make one dispel check, I can't think of any reason a caster would want to pick to dispel the same spell more than once, but, sure, go ahead, I guess.
