DCA seems to have a slight problem with unobstructed spotting areas. I saw on the design (published in the Post) before it opened that there was supposed to be an observation deck, but according to the people at the information desk, it does not exist right now.
Still, there are a couple of good places to look. If the approaches are coming from the north on the particular day (from the left, if you're not geographically inclined), go to either the window near the Great Steak and Potato Co., at the end of Terminal C. If you nudge yourself out of the way of the line, you'll have a nice view of the aircraft as they come gliding down past the Washington Monument.
Alternatively, you can go to the very end of the middle pier (considered to be Terminal B), and stand against the windows by the United gates. As long as you're out of the way, they won't bother you. However, the view is slightly obstructed, because you can't watch the aircraft roll out...just the touchdowns at the end of rwy 18. Same for the spotting location presented above.
If the approaches are coming from the south, you can go into the large central hall of the Historic Main Terminal (Terminal A), and sit in the rows of chairs that stand empty, waiting to be used. Once, long ago, this is where you could have seen DC-4s and DC-3s pulling up to the ramp, watching your friends and family coming down the stairs. Those aircraft are long gone, but these days, you can see jets as they roll out--the touchdown point is usually not visible, since the Main Terminal is not oriented in that direction. However, with the old second MT pier demolished, you get a very wide and expansive view of the airfield. Sometimes, too, Delta MD-88s and MD-90s are parked on the ramp, waiting to be flown back to Dallas and Atlanta. Be sure not to miss the airport museum at the top of the stairs, the picture display on the back wall, and the beautiful art deco design that is being restored to its original splendor.
Also, there are supposedly sitting areas on the top level with good views, accessed by going out the exists of the ticketing level and walking all the way to the right or left and turning toward the airfield. I have not visited these areas yet, so I can't confirm if they exist, but I'll be flying out of DCA next Monday, so when I'm there, I'll take a look and report back.
Hope that helps!