Roots hip hop is the name given to the original hip hop movement on the streets of New York. DJs like Cool Herc and Grandmaster Flash are credited among the musical pioneers that developed the new sound. They played party music, particularly funk and soul records, from the 1970s. They spun the records with a dance beat on a double turntable set up to create a new sound primarily for dancing at street parties and in clubs. Roots hip hop is specifically about the music, while the lyrical rhyming of rap would be added later.
Old school rap is the moniker that identifies the early style of lyrical hip hop music in the late 70s and early 80s. The style was developed as rappers or "MCs" began making up rhymes to go over the roots hip hop beats that DJs were spinning. The sound of old school rap evolved into less of a dance and party sound of groups like the Sugarhill Gang around 1983 by transitioning to a tough, urban street sound made popular by Run-DMC. Still, old school rap was a style of hip hop that focused as much on the lyrics as the beat, and the rhyming was simple, with the rhymes landing squarely on the beat without much variance that would appear much later in the genre.
One of the more controversial types of hip hop music emerged in the late 80s. It was known as gangsta rap. The term "gangsta" is a slang version of the word "gangster," and the lyrics of the hard-hitting style of music often involved the extreme violence sometimes found in city streets. Gangster and was sometimes accused of being displaying a disdain for women. NWA is generally recognized as the first gangsta group. Despite their songs, they were just entertainers and not the true gangsters that they pretended to be in the tracks. Still, the gang-related rap became very popular and survives today in many forms of modern hip hop.
Freestyling is a term that means two different things depending on whom you ask and what era of hip hop they are most familiar with. Freestyle used to simply be rapping prewritten rhymes over roots hip hop beats in the early days of the genre, but the more commonly accepted term for freestyle in the modern hip hop era is a type of improvisational rap. Simple beats back up the rappers who often compete head to head, making up rhymes on the spot in "battles" to show they are superior to the competition. The rap battles often involve insulting the skills of the other competitor. It is a type of hip hop found more commonly in live shows than on records.
Some call it an abomination of the genre, but the popularity of rap rock is a fusion of rock 'n' roll and hip hop that has stood the test of time. Bands like Linkin Park use rap lyrics in otherwise rock songs to create a sound that is popular with many fans. The idea of combining rock and rap started with Run-DMC and Aerosmith in the 1980s, and the idea was taken a step farther when Public Enemy paired with Anthrax in a metal rap version of the song "Bring the Noise."