The Yonge-University-Spadina line is one of Toronto's three subway lines, the other two being the Bloor-Danforth subway and the Sheppard subway. This line runs north-south along Yonge Street, makes a 180 degree turn at Union Station, and continues along University Avenue until it hits Bloor Street. After this, the line briefly runs along Spadina Avenue before branching off further to the west. In order to illustrate this, here is a rough map of the line, along with each of its stations:
                     ^                Finch
                     N           North York Centre 
  Downsview                          Sheppard . . . . . . .    
   Wilson                                |         
  Yorkdale                          York Mills
Lawrence West                        Lawrence
  Glencairn                              |
Eglinton West                        Eglinton
          \                         Davisville
       St. Clair West                St. Clair
            \                       Summerhill
           Dupont                    Rosedale
 . . . . . Spadina--St. George . . . . Bloor . . . . . . . .
                      Museum         Wellesley
                   Queen's Park       College
                    St. Patrick       Dundas
                      Osgoode          Queen
                    St. Andrew         King
                              \-Union-/

The Yonge-University-Spadina line intersects the Bloor-Danforth line at Spadina, St. George, and Bloor stations. Transfering at Spadina requires a bit of walking, which means St. George is usually faster, unless it's out of your way. This line also connects to the Sheppard subway line, which begins at Sheppard station and runs east.

The subway runs from 6:00 AM to 1:30 AM, except on Sundays (when the subway doesn't start running until 9:00 AM) and holidays. A generally useful page for schedule info can be found here.

Toronto landmarks that are located on the Yonge-University-Spadina subway line (and the stations closest to them):

Old City Hall/New City Hall/Nathan Philips Square/Osgoode Hall: Old City Hall is now a courthouse, but is still an interesting building to look at. Beside it is Nathan Philips Square and the new(er) City Hall building, which is also interesting, albeit in an entirely different way (try doing a Google image search for Toronto City Hall; you'll see what I mean). Bordering the square on the other side is Osgoode Hall, which has some really beautiful grounds (it's where my uncle had his wedding photos taken). The walk from Osgoode to Queen station (west to east) takes you past all of these, before arriving at the Eaton Centre.

The Eaton Centre: Connects to both Dundas and Queen stations, as well as the underground PATH walkway. Also somewhat interesting architecturally, it is still, at the end of the day, a mall. That said, much of the ceiling and walls are designed to let in lots of sunlight, there's a nice fountain in the middle, and there are famous geese hanging from the ceiling (no, really).

Union Station: A landmark in and of itself, with a great huge cavernous ceiling that I quite like.

The Royal Ontario Museum: Located at the aptly-named Museum station, this venerable stone building is about to have a huge crystalline mass grafted on to one end of it (if you didn't like the first addition, you're probably going to hate this one). Still a good museum, though.

Queen's Park: In addition to housing the Ontario Legislature, there is also an actual park here by this name, which divides the University of Toronto campus into its east and west components (getting from one end to the other in ten minutes is a challenge familiar to many students). While Queen's Park station has a direct link to the Legislature, the park itself is actually closer to Museum station.

Mount Pleasant Cemetery: You can see the cemetery from the subway when you're heading north to Davisville station. It's quite nice to look at, being at the crest of a hill with trees and groves and so forth, but as a destination in and of itself, I suppose it depends in large part on how much you would enjoy wandering around a cemetery. Alternately, if you take your bike with you, you could probably make it down to the foot of Yonge Street without pedaling once (well, if you got lucky with the traffic lights, anyway).