"Please don't use ambiguous speech."
Koumura Toshio, usually referred to as Koumura-sensei, is a recurring minor character in Clannad. He never has any episodes that delve into his past or into his character.
Koumura is an old teacher at the high school. His age means he is at times a little out of it, and he can get confused, but usually he's a kind gentleman who does his best to help others. Even though he gets confused, his memory can sometimes win out in a vital moment. His specialties seem to be working with delinquents and advising the choir club and drama clubs. His first appearance showcases these qualities well.
Chronologically, Koumura's first effect on the show would be at the beginning of Tomoya's high school career. As we learn in the flashback montage at the end of Sunohara's arc, Koumura was the teacher responsible for taking care of Tomoya and Sunohara (the two delinquents) and making sure they didn't ruin the school. That means Tomoya's always had a relationship with him, since his first days of high school.
Thus when Kouko wants to have her wedding at the high school, Tomoya goes to Koumura as the person who can make it happen. After a somewhat awkward conversation (due to Koumura's constant confusion about the identities of Kouko and her fiancee), Koumura agrees to get permission for the wedding. And it's a good thing he does, too: he ends up being the one to remind Tomoya and Nagisa about the wedding on the day of (they'd forgotten the wedding when they forgot Fuko).
Later on, when Nagisa needs an adviser for the drama club, Tomoya knows who to turn to: Koumura. When he tells them that the choir club has already asked him to be their adviser, the tension between the two clubs begins. After winning the basketball game, and thus winning the right to share Koumura as an adviser, the drama club is struck down by the student council. Eventually (thanks to, I'm guessing, some pressure from Tomoyo), the student council allows the sharing of Koumura, thus allowing the drama club to become an official club in time for the school festival.
The next time we see Koumura is at the first graduation of After Story, the graduation of Tomoya, Sunohara, Kyou, Ryou, and Kotomi. He tells them that he is retiring at the same time that they're graduating, and Tomoya and Sunohara bid him a fond farewell and thanks for all the help he has provided them over the past three years.
The final important instance in which Koumura appears is at the second graduation: Nagisa's graduation. After she is held back another year for her illness, and then being too ill to go to the official graduation, Tomoya and the others (including Kouko and Yuusuke this time) get Koumura to help them put on an unofficial graduation ceremony specifically for Nagisa. Koumura is the one who hands Nagisa her diploma after all these years (she had to be a senior for three years!).
Following the unofficial graduation, we don't see Koumura again until the final montage, where he sits on his porch, staring out into the world.
Effect on the Main Plot
Koumura is important as an administrator: they can be quite instrumental at times. Thanks to him, Kouko gets her dream wedding at the school, and, thanks to him, Nagisa and Tomoya remember in time to go to the wedding and have Fuko appear, thus allowing her to say her congratulations to her sister. Thanks to him, the drama club is (eventually) able to become an official club in time for Nagisa's dramatic scene at the festival (yes, drama and dramatic, it's a semi-pun). Thanks to him, Nagisa is able to have a graduation ceremony with all her friends (her family of friends). And even before all this, thanks to him Tomoya and Sunohara meet and become best friends, and who knows how Tomoya would have turned out without Sunohara? Thus Koumura's presence and help allows the events that help shape Tomoya and Nagisa's relationship (and also their lives) to occur, from meeting Sunohara to the graduation. Without him, their relationship (and thus the show) would be very different.
There's not too much involving themes with Koumura. The only thing I can really say is that Koumura is a vital part of Nagisa and Tomoya's extended family of friends and acquaintances. He shows that the vital and helpful nature of family is not restricted to biological family alone.
Koumura is definitely an important minor character. He has little screen time, and we never learn much about him, but without him many important moments in the show would not happen. He's a key force to the plot running smoothly. He also goes to show that old people aren't useless: they can be crucial allies in times of need. I also find it humorous that he's the one who remembers Kouko and Yuusuke's wedding when Nagisa and Tomoya forget when the earlier scene showed him completely spacing out on who Kouko even was!
That's all I have to say about that: Koumura is an old teacher who shows the usefulness of both teachers (and administrators in general) and the elderly.
Thanks for reading. God Bless, and peace.
Nota Bene: All clips are from the Clannad Central YouTube channel run by the Clannad (クラナド/Kuranado) fan page on Facebook. All character themes and other music from the show can also be found on said fan page, in the music player. My gratitude to them and all the work they do.