Salsa was most directly influenced by a genre called Cuban son, which was in fact a combination of Spanish and African music that was popular in Cuba with dance bands in the 1930s and 1940s. Son music layered Spanish guitar stylings over African rhythms to form a style that was widely popular throughout Cuba and even at Cuban dance halls in North America during the '40s and '50s.
Mambo was another Cuban style of music that had a strong jazz influence and was also very rhythmic, which made it good for dancing as well. Mambo bands kept the big band tradition alive during the bebop jazz era, which often saw bands shrinking down to tiny combos of three, four or five people. While not as big as big bands or some larger mambo bands, salsa bands eventually emerged to have quite a few members. In addition to the bigger band template, salsa would borrow many rhythms from mambo and its musical cousins, rumba and chachacha.
Soul could be considered black America's answer to rock and roll. It emerged during the '50s and '60s and became enormously popular worldwide through record labels like Motown and Stax. Soul groups like Smokey Robinson and the Miracles, Martha and the Vandellas and the Temptations shared the pop charts with Elvis, the Beatles and the Rolling Stones. In '60s and '70s NYC, soul influenced the Latino musicians who were busy fusing various musical elements into what would become known as salsa.
Bossa Nova was a Brazilian style of music popularized in the 1950s by such artists as Joao Gilberto and Antonio Carlos Jobim. Bossa Nova was known for its syncopated rhythms and slowed down samba beat epitomized in the Jobim classic "The Girl From Ipanema." Bossa Nova's sexy, seductive rhythms would prove to have a huge influence on the salsa music scene of the '60s and '70s.