Most tourists in D.C. look for accommodations close to the Smithsonian, and accordingly the East End is where most tourists wind up. There are lots of restaurants and nightlife options in the immediate area, you can walk to the Mall, and you'll feel like you're at the center of town.
But keep in mind that proximity to the Mall is really not so useful as proximity to a Metro stop. For a more authentic Washingtonian experience, visitors might prefer to stay in one of the numerous hotels just a little further north in Dupont Circle or Logan Circle, or just east in the historic Capitol Hill neighborhood. These neighborhoods are real hot spots among locals for their upscale dining and nightlife scenes. Moreover, you can actually find weekend street parking and avoid the $25-55 nightly fee hotels will charge you to keep your car downtown.
The West End also offers upscale hotels close to the Mall, catering especially to the business travelers who bustle along K St during the day. The downside to the West End is that the downtown commercial area is deserted after dark. A bit further west is Georgetown, which is perhaps D.C.'s most charming neighborhood, with a wealth of smaller, expensive hotels in the midst of a great dining and nightlife scene. Take note, though, that Georgetown lacks a Metro stop to keep out the riffraff, so you'll find yourself taking taxis or buses to get to the Mall and to other neighborhoods.
It's worth noting that Washington is a relatively small city, acreage-wise, and it's very easy and quick to stay in the close-in suburbs and take the Metro into town. You can save meaningful cash this way; suburban hotels are often substantially cheaper and D.C.'s hotel tax is an eye-popping 14.5%. Parts of Arlington and Alexandria, as well as Bethesda and Silver Spring, have easy Metro access into the District, and are worthwhile destinations in their own right.