"The day before yesterday, I saw a rabbit. Yesterday, a deer. Today, you."
Ichinose Kotomi is the subject of the second minor arc of Clannad: her arc runs between Episodes 10 and 14, roughly. After that, she exists mostly in the background, as a part of Tomoya and Nagisa's group of friends, and she never really becomes prominent again, except for one scene in the middle of After Story.
(Be prepared: this post is long! Also: split into two posts for easier reading.)
Kotomi is a high school senior (as are most of the school characters in Clannad), and she is basically a perfect example of a quiet genius girl. Her impressive intelligence is made apparent immediately, and it shows itself forth in numerous instances (such as when she bats). Her shyness is also very obvious. She spends most of her time in the library, before Tomoya drags her out into the sunlight, and she suspects everyone she meets of being a bully. Social skills are not her strong suit: she often uses strict, memorized phrases when first talking with a new person (like Nagisa, Kyou, Ryou, etc.), and even after that, her capacity for banter and casual conversation are somewhat limited. It's not hard for her to talk over people's heads, though. Kotomi has two passions: studying and playing violin. The former she is a master at; the latter is more to be classified as aural torture than music, usually (although good violin playing is present in her chamber orchestra theme, "Étude Pour les Petites Supercordes").
That's Kotomi in a nutshell: a quiet genius with a distinct lack of casual social skills and a passion for studying and playing violin.
Her social skills and violin playing are readily apparent in this scene:
We first meet Kotomi when Tomoya walks into the library and finds her sitting on the floor, surrounding by very academic books, about to take a pair of scissors to one of them. After an awkward conversation, Kotomi bids farewell to Tomoya, at least for the time being. Besides receiving a starfish from Fuko (and demonstrating impressive scientific knowledge about it), she does not really appear again until Episode 10, the episode following Fuko's arc. After a humorous incident in which Sunohara tries to ask Kotomi on a date and fails, Tomoya begins to talk with her, and he learns how she is basically friendless. Thus starts the quest of "Tomoya-kun" to gain friends for Kotomi. (She adds the -kun honorific to Tomoya's name. This honorific is used to show either closeness (such as between very good friends, childhood friends, or those dating/married) or an attitude of an elder speaking to a younger person (such as a teacher to a student, or a boss to an employee). Neither meaning is fit for someone your own age you've just recently met!)
Tomoya introduces Kotomi to all his friends, and, as she is somewhat forced into Nagisa's drama club, she begins to gain at least some social ability. That is, until she decides she can play the violin. Her skills cause aural pain to those around her, but her new friends are still kind enough to put on a recital for her.
The fun and games don't last long, though. Kotomi has a couple encounters with a mysterious man that she calls "the bad man." And following a scare in which the group of friends think Ryou has been seriously injured in a bus accident, Kotomi goes into a nervous breakdown and locks herself in her home. Meanwhile, "the bad man" continues to appear.
Tomoya and the gang learn that Kotomi's birthday is fast approaching, so they decide to help her celebrate. Nagisa, Kyou, and Ryou try to obtain a violin for Kotomi, but it is broken by a runaway driver and must be repaired. Meanwhile, Tomoya (with later help from the other three) works on renovating Kotomi's yard: pulling weeds, planting grass, rebuilding the flower bed, rearranging furniture. As he is working, Tomoya keeps getting a nagging feeling that the yard is familiar...and finally he has a revelation. Tomoya was Kotomi's childhood friend. He accidentally ran into her yard one day chasing butterflies, and they became friends. Everything was happy for a while, until Kotomi's birthday rolled around. Her parents had to leave on an urgent business trip, and Tomoya was supposed to bring friends to a party. Unfortunately, his friends didn't want to come to a girl's party, so he was too ashamed to go, and Kotomi was left alone...until the "bad man" showed up, that is. He was a work associate of Kotomi's parents, and he came to tell her that her parents had died in a plane crash, losing a valuable manuscript in the process. He wanted to look for an extra copy of that manuscript in the house. Kotomi panicked and found what she thought was the copy he wanted in her dad's study, and she quickly burned it. This started a large fire, though, that Tomoya (who eventually got up the courage to visit her alone) and the bad man came and put out for her. Kotomi then learns that what she burned was not the manuscript, but a catalog her parents were looking through to find a teddy bear for her birthday. After that, she truly became a social recluse.
(See Kotomi's side of the story here, and Tomoya's side here.)
Jump back to the present. Tomoya and the others finally convince Kotomi to leave her house, and they force her to listen to what the bad man has to say. He's brought a briefcase that was her parents'. It turns out this briefcase survived the crash and travelled around the world to her (in either a miraculous or cynically impossible way). Inside is a teddy bear she requested for her birthday and a letter from her parents. Kotomi's joy is incandescent as she receives that final message from her parents.
Thus, after her birthday party, ends Kotomi's arc. After that, she is mainly a background character. She has her humorous moments, like all background characters, but she doesn't really have any major moments, except for during the New Year's party in After Story. When all the friends visit Tomoya and Nagisa for New Year's, Kotomi explains her continuation of her parent's work (detailed below in the "Effect on the Main Plot" section) while she has been in America (where she went to study after high school). The final shot of Kotomi is her wearing sunglasses driving a fast convertible down a winding highway, her hair blowing in the wind.