Story Azumanga Daioh is a show about nothing, literally. Think of it like Seinfeld, but with high school girls and more hilarity, and you've got all the bases covered. The episodes revolve around six girls and their daily life at a high school, which always ends up being random in some way.
The comic formula is hilarious, and is almost flawless. Many scenes will make you scratch your head and wonder "eh?" while simultaneously laughing, while other scenes make you laugh by repeating the same thing over and over and over and over, far past the point where you think it should have just stopped the joke. The best comic relief was definitely the male teacher who was always hitting on the girls, who had a trademark jaw drop constantly. He would randomly be thrown into the episodes very spasticly, making anyone with a sense of humor laugh out loud. I think that's the beauty of Azumanga Daioh: it relentlessly attempts at making you laugh by throwing you off guard by whatever means necessary, and it succeeds. So really, don't go into watching Azumanga thinking you'll get any sort of a plot. There are only a few episodes that are linked somehow, but the rest do follow a timeline (the girls making their way through three years of school, up to the satisfying conclusion).
My only minor complaint would be the same as any random humor show: they never seem to maintain a constant hilariousness the entire way through. I don't expect this by any means, but I also always would prefer a shorter show with nothing but hilarity, as opposed to 26 episodes dragged out with some episodes that just don't cut it in my book. Granted, Azumanga has the highest percentage of purely funny stuff to generic laughs than most comedic series I've seen (others include things like Haunted Junction, Abenobashi, etc), but it still would have been nice to keep the same level of humor the entire time. Because of the randomness, as well, I don't think this is a series to watch all in a row, unless you are doing a marathon and are really in the mood for a comedic burst. Otherwise, watch a few episodes once in awhile, intermixed with a different show you also are watching. This keeps the humor fresh and doesn't burn you out. Animation Azumanga has a very minimal style of animation, with not a lot of detail or shading, but hey, it doesn't really need it for the content involved. Randomness and humor abound, so there are a great deal of chibis and deformed scenes that take place. The spastic nature of the show is reflected in the spastic animation and color schemes, and it works very well. The emotions of the characters are drastically exaggerated by using huge eyes or ridiculously overdone expressions, and this helps the humor a great deal. There isn't much else I can say about the animation besides this. There is no CG used and no crazy special effects, just minimal shading and artwork, and tons of crazy scenes. Sound The music in Azumanga is quirky and light, just like the tone of the show. Sometimes band instruments are used such as flutes or drums, which help to amplify the bizarreness of the scenes. Quite frankly, the music was perfectly placed with the tone and mood of the series, but I still deducted a few points because the music did get a bit repetitive over time. No new songs were introduced and the same few songs were played over and over. Yes, repetition was the point of the humor at times, but the music could have had a face lift once in awhile for good measure. The outro song was one of the catchiest I've heard and I'm not sure why. Voices were great for everyone involved, and really helped as far as the exaggerations for some of them (Osaka and Chiyo, for example). Characters The character development was basically non-existant, yet I still gave this section a 9.5. Why? To put it bluntly, Azumanga Daioh has some of the most hilarious characters I've seen, who all fill their incredibly exaggerated role perfectly. Chiyo is the little girl who is the epitome of cuteness. Her voice helps with this persona, as well as giant watery eyes when she gets upset, and her little girl ways of doing things. Sakaki is obsessed with cute things and is super quiet, and always blushes whenever she sees a cat or dog, yet (for the comedic effect) she always is bitten when she tries to pet things. Tomo is the most obnoxious character by far and is always yelling and pissing people off. She's why I deducted the .5, as she was irritating as hell. Yumi is probably the least developed character -- her one quirk is that she is obsessed with her weight. Then there's Osaka, an extremely slow and stupid girl who tends to always make you laugh with the idiotic things that come out of her mouth. And finally, Kagura is athletic and competitive. These six characters are fairly extreme to begin with, but them morph into super extreme characters by the end of the series. Osaka seems slow at first, but becomes almost braindead by the end of the series, for example. It's like watching old Simpsons versus new Simpsons... Homer is way more defined and hilarious in the later series than the first, and Azumanga is the same. The entire series is about pushing the limits of comedy, and the characters help out with this immensely.
More characters are on the front lines than these six, though, such as the forgettable Kasumi (who has a weird crush on Sakaki), the two rival teachers (Yukari-sensei being the violent and drunk of the two), and the guy who steals the show: the male teacher obsessed with the high school girls. His jaw is always dropped and he's always coming up with creepy and inappropriate things to say, and that makes him all the more hilarious! Literally every scene he was in made me laugh out loud, and that's a feat for any series.
Overall Azumanga is a great watch for any fans of comedy. Out of all the random series I've seen, Azumanga definitely has the highest concentration of randomness and laugh out loud moments, combined with hilarious characters and wacky moments alike. If you like comedy series, I can't see why you haven't watched this yet! And if you haven't, pick it up right now, since you'll surely like it. Two thumbs up in my book.