how can i make it so the camera rotates around an object and continually faces it?
switch to top view (numpad7)
create a bezier-circle. exit edit mode
parent the camera to that circle, choose “follow path” (or however it’s called)
press F7, you’ll see the constraint tabs and a follow path constraint assigned to the camera (make sure to select the camera itself to see the settings)
click “curve follow”
in the 3d viewport press ALT+G, ALT+R (clear location, rotation of the camera to reset its transformation), then press “I”>> LocRot
in the constraint panel: for Fw choose -X, for Up Y.
press ALT+A to play the animation
hope that’s it 
why not:
add an empty at the center of the object
position the camera pointing at the empty
make the empty the parent of the camera [select the camera, then [shift+rmb] select the empty, press control+p]
animate the empty spinning, it will probably be easiest to add two keyframes and then go into the IPO window and make the curve a line [v key in edit mode for the curve], then set the extend mode to extrapolation [it has moved into the menus somewhere]
no! no! no!
just add an empty to the center of the object then select camera than empty and press cntrl + t and make the camera track the empty. Now key frame the camera to rotate around the empty and the camera will self align with the empty (the center of the object)
turn mblur on and you can creat a cool spinning effect!
i think that is alot easier although it too has it’s limitations! 
this is good for free-form motions, a circular motion needs some tweaking of the IPO curves. i agrees zerod’s method is the easiest for that particular task