Brain Fertilizer

As always, this is my assessment of who is influential, not necessarily who I like, or who I think is the most skilled (although you have to be particularly skilled to make this list). These are the drummers who I know by name, whose style I can identify without hearing the song with it, the ones whose style is a point of comparison in music reviews, and/or the people I have heard cited by drummers as people they want to drum like.

1) Buddy Rich
Even though he's a jazz drummer, his ability and style influenced all drummers. He is, quite simply, the best ever.
From his wiki entry:

Rich's technique has been one of the most standardized and coveted in drumming. His dexterity, speed and smooth execution have been considered Holy Grails. While Rich typically held his sticks using traditional grip, he was also a skilled "match grip" player, and was one of few drummers to master the one-handed roll on both hands. Some of his more spectacular moves are crossover riffs, where he would criss-cross his arms from one drum to another, sometimes over the arm, and even under the arm at great speed.

He often used contrasting techniques to keep long drum solos from getting mundane. Aside from his energetic explosive displays, he would go into quieter passages. One passage he would use in most solos starts with a simple single-stroke roll on the snare picking up speed and power, then slowly moving his sticks closer to the rim as he gets quieter and then eventually playing on just the rim itself while still maintaining speed. Then he would reverse the effect and slowly move towards the center of the snare while increasing power.

Another technique that few drummers have been able to perfect is the stick-trick where he does a fast roll just by slapping his two sticks together in a circular motion.

2) John Bonham
Great drummer. Perhaps a little messy, but great passion, groove, and ultimate feel. I once heard him described as "the loudest drummer ever."
From his wiki entry:

He was renowned for his power, speed and "feel" for the groove. Bonham is described by the Encyclopædia Britannica as "the perfect model for all hard rock drummers that have followed him".[

3) Neil Peart
In contrast to Bonham, Peart is one of the most precise and cleanest rock drummers I've ever heard. From his wiki entry:

Peart is often regarded as one of the finest practitioners of the in-concert drum solo. He is known for extensive, intricate drum solos containing odd time signatures, complex arrangements (sometimes total separation between upper and lower limb patterns.

4) Lars Ulrich
Thrash Metal is to drumming what Neo-Classical Yngwie J. Malmsteen-style speed soloing is to guitar: even if you don't play that style, the technique has to be part of your repertoire. And Lars Ulrich spearheaded the effort. From his wiki page:

Ulrich's early work with Metallica led him to be dubbed as one of the thrash metal scene's drumming greats. He became known as a pioneer of fast thrash drum beats and techniques, featured on many of Metallica's early songs... He has since been considerably influential due to both the popularity of his band and his interesting drum techniques, such as the machine-gun double bass in the song "One" and the pounding opening of "Enter Sandman".

5) Keith Moon
I don't know that much about him, but he is clearly one of the most influential rock drummers of all time:

Moon is known for innovative, dramatic drumming, often eschewing basic back beats for a fluid, extremely busy technique focused on fast, cascading rolls across the toms and cymbal crashes. Moon was one of the first to play the drums as a lead instrument in an era when drums were supposed to only keep the back beat. He is widely regarded as one of the greatest and most distinctive rock drummers of all time.

6) Ginger Baker
I'd heard that the members of Cream were all egotists trying to upstage each other. If so, I think that despite Eric Clapton getting all the fame*, Ginger Baker was actually the most influential member of the band. According to his wiki entry, Ginger was in the same flamboyant mold as some of the other influential drummers in this list, but spearheaded the use of a double bass drum. That's worthy of inclusion right there.

7) Stewart Copeland
Just slightly less crisp and precise than Peart, Copeland still hewed his own path to a distinct enough sound and style that he gets his own mention. From his wiki entry:

Copeland is known for his precise, energetic, and creative rock drumming along with a reggae and jazz influenced style. His distinctive sound centers on a hard, high-pitched crack on a snare drum or rimshot, subtle hi-hat work with understated flourishes, while often playing only hi-hat with bass drum. Copeland is a master of the syncopated beat, and his distinct approach consolidates his position as an important drummer on the world stage, subsequently influencing generations of drummers.

Copeland is also noted for his heavy emphasis on the groove as a complement to the song, rather than displays of technical prowess. He once drove this point home at a drum clinic: Copeland announced that he would show the audience something "that very few modern drummers can do," and proceeded to play a simple rock beat for two minutes. Nonetheless, his playing often incorporates spectacular fills and subtle inflections which greatly augment the groove. Compared to most of his 1980s contemporaries, Copeland's snare sound was very bright and cutting. Another novelty was his use of splash cymbals. He also is one of the few rock drummers using the traditional grip rather than the matched grip.

8) Rick Allen
His wiki entry sells him short, in my opinion.
Yes, Rick Allen is influential because he's probably the only one-armed drummer in the world, ever. But he earns a spot on this list because he is one of the best drummers I've heard at surprising sequences of beats. He doesn't do anything that complicated, but it never fails to sound special to me. I might consider him the equivalent to Carlos Santana: his musical sense is impeccable, even if his technique is nothing to write home about.

Honorable mentions:
Cozy Powell
Carl Palmer
Carmen Appice
Ian Paice
Aynsley Dunbar
Bill Bruford

Not making the list:
Phil Collins
Alex Van Halen
Tommy Lee

*despite Mr. Clapton being famous for his purported guitar skill, I think he actually is known for his singing and songwriting while simultaneously being a slightly-above-average blues guitarist