Gags

Tsumiki's "Secret" Crush on Io
The central core of the story revolves around Tsumiki's unrequited crush on classmate Io Otonashi. They share that grayzone area of "more than friends, less than lovers", which Tsumiki desperately hovers in and out of. This is made EXTREMELY difficult due to the fact that she is a natural "tsundere" character - where she says one thing, and means something else - a fact that she is often teased about by her friends. Most of the gags centered on Tsumiki stem from her relationship and feelings towards Io, whether directly or indirectly.

Sakaki Attracting "Flying" Objects
In almost every episode of the anime Sakaki has been shown to attract "flying" objects that injure him in various ways- many times these objects aren't even aimed at him. These objects include frisbee (ep.5), a sign (ep.9), part of an umbrella (ep.6), a Mayoi "Super Headbutt" (ep.10), and even multiple fireworks (ep.7). The only exceptions to this are either when he is throwing the object or the target is Mayoi.

Kyoya's "Secret" Crush on Saki
Kyoya's crush on Saki parallels Tsumiki's crush on Io in various ways. However, Kyoya has a much more open relationship with Saki than Tsumiki does with Io, as they are often seen partnering up or hanging out together.
 * Everyone except the recipient of their feelings know about their crush.
 * Both are "tsundere" either when addressing their crush or talking (denying it) about it to others.
 * They get easily embarrassed when receiving a gift or compliment from their crush.

The "Score"
Throughout the series, the gang has held many different competitions between each other, with there always being winners and losers.
 * Io and Sakaki have the fiercest friendly rivalry, which seems to have lasted as long as they have known each other. That being said, Sakaki has won only twice in the series, both times against Mayoi who hasn't won any competition.
 * Io has only lost once in the series, this is attributed to his teammate Mayoi (who has never won a competition) quite literally falling short of the goal.
 * Mayoi has lost every competition she has participated in, even in the ones she made up.

Mayoi's Inventions
Throughout the series Mayoi has made many different inventions that are mostly used to carry out her pranks or be used in their competitions, none have very practical uses otherwise.
 * Yukidama Compression Bazooka
 * Snowball Replenisher
 * Wrist Sensors
 * Hydraulic Lift for Kick the Can Tournament
 * Bear Costume with mask

The Hairdos and Getups
Each of the characters have their own distinct hairstyles and outfits, which can be somewhat symbolic.

Tsumiki Io Sakaki Mayoi Hime Saki Kana Miiko
 * Tsumiki is often seen "growing" cat ears, particularly when embarrassed or annoyed. There are many cat related themes throughout the anime that are often referenced back to Tsumiki.
 * Io always wears glasses except when they are taken or when he's asleep. In the OVA his glasses are used as a good luck charm, which gains Kana many cans of soda. They also emphasize his studious and more serious nature.
 * Blond hair is often used for the delinquent or rowdy characters in anime, and Sakaki is no exception. He is the witty best friend who often is the butt of everyone else's jokes.
 * Mayoi is almost always seen wearing her lab coat to school with the exception of the Founder's Festival when she wore the Tsutsumin stand uniform. Interestingly, while she wears the lab coat so often that her classmates point it out when she doesn't wear it, she is almost never seen wearing it at any other time.
 * Fun fact; Mayoi doesn't have eyes. Her bangs always overhang her eyes, and so her eyes are never actually seen in the series. When her eyes are depicted it is usually with the "eye gleam" effect, but never actually showing her real eyes.
 * Mayoi's two-bun hairdo makes her stand out instantly, and is often referenced to different kinds of animals, most notably a frog or bear. Another way of looking at them is to say that they are tanuki (raccoon dog) ears, as they are often shown to be mischievous, which more closely matches Mayoi's personality than they other animals.
 * Hime is kind of depicted as the normal (personality-wise), pretty girl of the group, and so she is often seen with innocent-looking outfits (matching her naivette).
 * Saki's hairdo is often referred to as cat-like, but to me they make her look a lot more like the kitsune/foxgirl characters in anime, which makes sense due to the way she is "gracefully mischievous".
 * Kana's double ponytails make her appear childish, and indeed, she is the most immature of the group.
 * Miiko's hairdo is kind of reminiscent of dogs. Her "aura" that makes it impossible for Io to refuse her is similar to characters succumbing to girl's "puppydog eyes".
 * Miiko has never opened her eyes in the series.

Sakaki's "Teleportation Jutsu!"
Several times throughout the anime, Sakaki appears seemingly out of nowhere (usually leaning in the doorway) as Io puts it, at just the right time. This mostly occurs when they are about to start their antics for the day, or when he must excuse himself from said antics.