Takeshita Dori (竹下通り) (sometimes called Takeshita Street or just Takeshita) is the famous Harajuku shopping street that begins just outside of the "Takeshita Exit" of Harajuku Station on the JR Yamanote Line.
The most well known (and crowded) part of Takeshita is the segment that runs between Harajuku Station and Meiji Dori. Further from the station, on the other side of Meiji Dori, Takeshita continues on for several more blocks, intersecting Cat Street, continuing on and finally ending at a perpendicular street that runs along one end of Omotesando Hills.
Many of the shops on the Harajuku Station to Meiji Dori segment of Takeshita target Japanese teen girls and are famous for offering fashion catering some of Japan's more interesting subcultures, including lolita, punk, gothic, and hip hop. There are also numerous crepe shops, restaurants, a few international brand shops, ticket sellers, and other types of businesses. This part of the street is often packed (especially on weekends) with thousands of colorfully dressed Japanese junior high school students and is a very popular people watching area for tourists and locals. Just off of Takeshita, and running parallel to it is another small (less popular) street packed with cafes and shops called Brahms Path.
The segment of Takeshita on the other side of Meiji Dori (between the KDDI Design Studio and the end of the street near Omotesando Hills) has a more relaxed feel but is also dense on both sides with fashionable shops. This segment of Takeshita seems to have more clothing stores aiming for young well-dressed Japanese guys than the Harajuku Station side.
On the map below, the blue line to the right of Harajuku Station is Takeshita Dori.