Alphabet City is in the East Village
Cooper Union is an academic institution as well as an exhibition space. They use the money from their exhibits to subsidize their students' education. All students who attend Cooper Union have a FULL SCHOLARSHIP, so you could imagine how competitive that school must be. They offer only three degrees: Bachelor of Architecture, Bachelor of Fine Arts, and Bachelor of Engineering. For more information, log onto: http://www.cooper.edu/. They also have a library, which I have yet to visit. I heard there permanent design collection is extraordinary.
I guess this would be self-explanatory, but the reason why this section is called "Alphabet City" is because most of the streets are one-letter streets (e.g. A Street). St. Mark's Street is a cool street. They have unique shops and restaurants. They have a wood store, but not like Home Depot sells lumber, it's very high-end.
I heard the "2nd Ave Deli" is good. The type they used in their signage looks Hebrew. . .pretty cool.