Seoul Station is the main departure point for all points Korea in Seoul. It's not the only one, though - Yangsan Station has high speed rail trains down the west coast, and there are plenty of bus terminals and other major stations in town if you're heading to other places in Korea.
The building has a Byzantine-style dome and brick facade that reminds me of somewhere particular, but I can't remember exactly. I believe it's a university or museum of some sort, though I'm sure there are probably dozens or even hundreds of buildings that have a similar style. There are probably buildings just like this in Tokyo that I just can't recall!
Anyway, the old Seoul Station still stands. Renovations were completed in 2011 and it now serves as a cultural center; the new building is just south of the old one, and much larger, modernized and designed to handle the KTX high speed rail lines. At nearly 100 years old, it's worth seeing if you're a history or architecture fan, though getting inside will probably require attending an event; for what I understand only the shell is preserved.