(Super) Azusa is a limited express connecting Shinjuku station in Tokyo and Matsumoto station in Nagano(there is a daily train extending to Minami-Kotari). The frequency is quite high and has long train composition.
It is one of the busiest limited expresses excluding Shinkansen. All Azusa call at Hachioji station(main station of second largest city in Tokyo), Kofu station(main station of a prefectural capital of Yamanashi prefecture), Chino station and Kami-Suwa station in Nagano. There four types of passengers on board. First one is local people traveling to Tokyo. Second is businessmen/women. Third one is tourists and the last one is hikers.
Especially passengers heading to Matsumoto from Tokyo are mostly hikers or want to enjoy the nature around Matsumoto, for example Kamikochi district, rather than having sightseeing in Matsumoto.
There is another limited express running between Shinjuku and Kofu, which is called Kaiji. Kaiji stops more stations than Azusa, and especially at Otsuki station which you can change to Fujikyuko line to Kawaguchi lake.
Azusa is also convenient for tourists who want to go to Tsumago in Nagano from Tokyo, rather than taking Tokaido Shinkansen to Nagoya and change to limited express Shinano. Take the Azusa which stops at Shiojiri and take Shinano bound for Nagoya and get off at Nagiso station.