Her guitar is a Fender Japan '69 Reissue Mustang in Candy Apple Red, specifically the version with matching headstock color; the matching headstock color is a trait exclusive to this particular model, only made by Fender Japan. After Yui named both her own and Mio's guitars, Azusa named her own Muttan, (taking the first syllable, like Ritsu/Ri-chan; "-tan" being an alternate of "-chan") as Muttan sounds like "Mustang". She does not have the Tremolo Arm (commonly known as a Whammy Bar) installed, though it comes standard with the guitar. 
Her surname is taken from former P-Model keyboardist/programmer Teruo Nakano
The kanji of her given name (梓), when read "azusa" as in her name, refers to either of two tree species: the Japanese cherry birch and the yellow catalpa.
The other girls sometimes comment on how much Azusa and Mio resemble each other (especially during the few occasions when their hair is styled the same way) to the point where they could pass for sisters.
Both Ui and Jun mistake Azusa for a stranger when they see her tanned for the first time. This happens twice with Ui, despite the fact that she was visiting Azusa at home and had seen the tan once before.
Though she claims to dislike being treated like a cat or a child, Azusa's behavior is often considered cute by the other girls (and indeed by many fans of the series) and this causes the other girls to treat her that way. Examples of her "cute behavior" include pouting when no one is willing to practice, and meowing like a cat when startled or sad, and forgetting she had animal ears on overnight.
Her parents used to be in a jazz band, and this musical influence encouraged her to become a musician herself.
Jun and Ui have described her as an anime figurine on at least two occasions.
Azusa loves sweets, especially cake, but tries to hide the fact that it's one of her weaknesses.
Azusa's nickname, "Azu-nyan", was given to her by Yui after trying on a pair of cat ears. The nickname eventually became so popular for Azusa that real-life fans refer to her with that nickname possibly as much as her real name.
Azusa is considered by many fans to be the most child-like in appearance of the characters, but is the most mature and serious in personality alongside Mio.
Much like how her seniors can be regarded as female counterparts to The Beatles, Azusa herself can be regarded as the K-On! universe's counterpart of Yoko Ono. Like Ono, Azusa's skill and artistic talent did much to influence the style and performance of her band during the duration of her membership, not to mention she has the open and almost constant affection of the K-ON! universe's John Lennon, Yui. However, in a heartwarming inversion of Yoko Ono, rather than breaking the fellowship of Ho-kago Tea Time into tatters of hatred the way Ono did with The Beatles, their collective love for her brought their sisterhood and friendship closer together by the time the fellowship disbanded with the founding four members' graduation.
In the anime, Azusa has her very own, cat-themed tea-cup which she got from her seniors. In the manga, she is using a normal tea-cup from Mugi's tea set.
In the anime, all members of Ho-kago Tea Time have matching key-chains on their bags. In the manga, she is using a plectrum, that she got as a souvenir from Yui after the Kyoto field trip.
Azusa proves a common misconception wrong twice in the second season. Most people assume that someone who's a good guitar player is also a good guitar teacher. While proficiency on the instrument is a huge asset for any teacher, proficiency alone is not enough. Azusa proves this by first trying to teach Ritsu to play "Fuwa Fuwa Time", by having the sheet music for the vocals (probably an error) in front of them, despite Ritsu not knowing how to sight read. The second time, she tries to teach Mugi to play guitar, starting with the F Major chord, saying it causes a lot of beginners frustration (which is correct). Normally, beginners don't learn barre chords such as B and F Major until they have all the 8 basic open chords down.
In the first season's ending theme, "Don't say "lazy"", Azusa is the only one of the group to not be present after her debut episode. This is in contrast to the first season's opening theme, "Cagayake! GIRLS", which is updated to include her along the group as well as the instrumentals of her rhythm guitar straight after her debut episode.
Oddly enough, "Don't say "lazy"" has an "enhanced" version which includes the instrumentals of her rhythm guitar but was never used in the anime.