This is a very cursery overview of jazz and should be now means be thought of as diffinitive. There a many many great resources out there if you're interested in learning more about the music and it's origins.
Many people consider jazz to be uniquely American music. New Orleans is often credited with being the birthplace of jazz. The port city at the end of the 19th century was an incredible mix of cultures and people bringing with them their own musical traditions. From the beginning, African American musicians were the core innovators in this new style of music. As musicians began to combine the traditions and music of the Blues, popular marching band tunes and a spirit of collective improvisation the early era of jazz began. The most influential innovator to emerge from this New Orleans style was Louis Armstrong. To many, he is the father of modern jazz.
The next major era of jazz occured as the music moved out of New Orleans and into cities like Chicago, St. Louis and New York. Beginning in the 1930's and continuing through the 40's, jazz entered into what has come to be its most popular and accessible time - the Swing Era. Characterized by big bands led by musicians like Count Basie, Benny Goodman and Duke Ellington, the swing era was the period when jazz was the most popular music in America. Groups of twelve to sixteen members toured the country playing packed dance halls. As the bands got bigger, less emphasis was placed on collective improvisation and more on the musical arrangements.
Following the swing era, jazz entered into one of the most important creative eras since the New Orleans style began. As the large bands of the swing era began to disband for a variety of reasons, small groups emerged as the hot bed for jazz innovation. In this period from the mid 40's - mid 50's Bebop was the reigning style of jazz. Pioneered by musicians such as Charlie Parker, Dizzie Gilespie and Thelonios Monk, bebop was a new sound for jazz. The pace was quicker and complex harmonic structures were combined with popular melodies to create a new music. With bands composed of only four to six members, musicians had more space to experiment with improvisation. Musicians who would come in their own in the sixties, like Miles Davis and John Coltrane.
The next major creative step for jazz began in the sixties with the emergence of an avant garde movement. Building on such work as Miles Davis' Kind of Blue of 1959 and then Ornette Coleman's Free Jazz of 1960, jazz musicians began to take the music to the boundaries of creativity. As the sixties progressed, innovators like John Coletrane recorded A Love Supreme and live performances at the Village Vanguard in New York. Others like Cecil Taylor and Albert Ayler played some of the most challenging music to date. Jazz musicians like much artists of the era also began to speak out on social issues and used their music as a vehicle for political and social expression. Musicians like Archie Schepp and Charles Mingus were both prominent in this regard.
The current era of jazz could be said to be a combination of the avant garde and a neoclassical movement. Probably the most famous contemporary jazz musician is Wynton Marsalis. His precise compositions incorporate many elements of the New Orleans and Swing styles while also using innovative techniques. Other musicians like Greg Osby and Steve Coleman incorporate more avant garde styles into their music. Other musicians continue to push the boundaries of the music by incorporating Hip-Hop with free, collective improvisation.