"Onsen" originally means "hot spring" - in extension, the word is often used to refer to the hot spring bathing facilities around such springs, or the traditional inns that use such spring water for their baths, or even to the whole resort town that is renowned for the local hot spring. Many towns referred to in this way are rural towns - due to the natural consequence of them being close to mountains with volcanic activities. People go to those resort places to enjoy the hot spring bath, natural scenery, and often the outdoor bath there - for relaxing, so people *tend* to stay over, but not always.Also, spa resorts that use such "hot spring" water is called that too.
If you want to just try spa resort facilities in central Tokyo, I believe that Spa Laqua, near Suidobashi (JR) or Korakuen (Metro Marunouchi line) station, is built using the natural hot spring water:
http://www.laqua.jp/spa
So it Oedo Onsen Monogatari in Odaiba (spa theme park, but uses natural hot spring):
http://www.ooedoonsen.jp/higaeri/english/index.html
Another one in Hakone, if you are making a day trip from Tokyo, would be Yunessun:
http://www.yunessun.com/english/
If you don't want to go to a whole "spa resort" facility but want to enjoy it the "minimum fuss" way, that would be called "day visit" (higaeri) onsen, as against staying over. Some inns with hot spring bathwater do provide such day visit service.
