School
The school that Lain attends along with her classmates Alice, Reika and Juri has no name in any of the works. Alice's teacher and Lain's teacher work here. According to Visual Experiments Lain, it's called Ouka Private Girls' Academy (私立鷗華女子学園, shiritsu ouka joshi gakuen, but I might have the pronunciation of ouka wrong).
Description
The school buildings are mostly white. There is a main gate all the students enter through, beside which there is a bust of a man (unidentified, presumably the school founder). In the centre, past the gate, is what appears to be a large pattern.
In the classrooms, students are seated at simple individual desks facing the blackboard at the front of the classroom. There are large windows lining the outside wall and the wall facing the corridor.