Well, that depends on how you define "best."
Most Seattlites think the best neighborhoods are those that have great views of the water -- either one of the lakes or Elliott Bay. Views of the mountains are good, too. When you have views of both water AND mountains, that's as good as it gets in Seattle.
The Seattle neighborhoods where you can get a house or condo with both water and mountain views are along Elliott Bay (Magnolia, West Seattle) and along the western shore of Lake Washington (Leschi, Madrona, Madison Park, Laurelhurst).
Another option is to use height/altitude as your guide. In nearly every neighborhood, there's a high hill where you have good views of water and/or mountains. (The exception is the Central District, which has NO views and is NOT a very nice neighborhood.) Even in the downtown core, if you get a condo on a high-enough floor on the "good" side of the building, you'll have great views.
On the other hand, if you have a family, you might define "best" according to things such as schools, parks, access to shopping, etc. And then there's easy commuting to downtown (if you're going to work downtown). For those things, I think Queen Anne is great -- it has all of those things. Also Magnolia, which is a bit farther from downtown but not too far. Queen Anne and Magnolia are both very nice neighborhoods. I think #3 is Madison Park. about 7 years ago