Asakusa Town

Asakusa is an old town with some classical flavor. Asakusa is one of good choices for a town to stay in Tokyo. It has a laid back and lively old quarter atmosphere, right in the central part of Tokyo. And it has easy access to most tourist spots in Tokyho.

Transport

There is a direct train from Narita Airport to Asakusa - Keisei Main Line
There is a direct train from Haneda Airport to Asakusa - Keikyu Line
Two subway lines to the center pass through Asakusa - Ginza Line and Asakusa Line (Asakusa Line operates seamlessly with both Keisei Line and Keikyu Line.)

To See

See gorgeous Sensoji Temple (Bhuddist) and Asakusa Shrine (Shinto) at one place!
Enjoy the quite lively 'sando' (road leading to a temple) which offers so many shops, food and genuinely Japanese souvenirs.
Stroll along the traditional malls and feel the down-to-earth atmosphere.

Hotspring in Asakusa!

'Jakotsu-yu' has been around since Edo era.
Address: 1-11-11 Asakusa, Taitao-ku, Tokyo
Brin your own hand-towel and shampoo. Closed every Tuesday. Price: 460 yen.
Official site (in Japanese): Jakotsuyu

Boat to Odaiba

Hop on a water bus to Odaiba and enjoy the great view! The boat from Asakusa goes down the Sumida River, and then continues to Odaiba through the Tokyo Bay.

Where to Stay in Tokyo

You may be asking, "Where should I stay in Tokyo?" There are a few good options, but Asakusa town is probably the best for many people.
Asakusa is an 'old town' in Tokyo. Asakusa itself is in a central location and has especially good access to most tourist destinations with two metro lines, two train lines and a boat for Odaiba. Asakusa Metro station is served by two subway lines:

Asakusa line - directly connect Narita Airport, Asakusa, Shimbashi, Shinagawa, Yokohama and Yokosuka, even without transfer. Also easy transfer to Roppongi - the most popular bar/club area filled with executives, celebrities and expats at night. (Change at Daimon for Toei Shinjuku Line.) For Akihabara, transfer to JR Sobu Line after a few stops, at Asakusabashi station.
Ginza Line - directly connects Asakusa, Ginza, Akasaka, and Shibuya. The Tobu line runs directly from Asakusa to Nikko, a scenic tourist spot in the north of Kanto Area.

But most of all, I can recommend Asakusa for its overall 'atmosphere' which is very different from busy business areas or quiet residential areas. That is probably why Asakusa attracts so many tourists every year.
Asakusa has many hotels and hostels, many of which are best rated in Tokyo. You can also find some 'Ryokan' style hotels with old-fashioned 'tatami' floor.
And here are a few more things to note:

You'll find a lot of traditional food and goods, especially around 'Sensoji'.
Asakusa has plenty of hotels and hostels, as well as classic style 'Ryokan's.
Somehow, backpackers in Tokyo seem to like Asakusa.
You can visit Odaiba island by direct boat from Asakusa. It takes less than an hour.
Most of all, you can JUST WALK to everywhere in Asakusa from your hotel.