Oh I studied this at university! In terms of the development of rock 'n' roll, which itself went on to spawn many different genres, jazz, swing, and rhythm and blues were all extremely important. It's a bit tricky to grasp just how much pop music owes to these styles, because we're so used to how they sound and they've been developed over the past sixty or so years anyway, but I'll highlight a few crucial things. It's easiest to relate jazz to early pop music, so I'll do that and then you can infer how early pop developed ino contemporary popular music and which bits have been changed or adapted.

Harmonies

Harmony is about what chords you use, and the melodies (tunes) you put over the top (such as vocal melodies or guitar solos). You probably know the twelve-bar blues - all it is is a series of chords that have been used to structure hundreds of songs, probably developed at some point in the early 20th century and most used in blues music (hence the name). It's also used a lot in early rock and roll: 'Shake, Rattle 'n' Roll', 'Rock Around The Clock' and the 'Hippy Hippy Shake' are big hits that use it really clearly.

The beauty of the twelve-bar blues is that it's simple - you only need to know three chords - as well as easy to improvise over, and easy to play around with. Early Beatles music is often based on the twelve-bar blues, as is some of the Rolling Stones.

A whole load of other jazz progressions exist, but the twelve-bar blues is far and away the most important. You also get a huge number of other harmonic tricks making their way from jazz to pop, including some 'jazz chords'. Without getting too technical, these are like 'normal' chords with extra notes added to give them flavour, and occur all over 'Taxman' by the Beatles if you want an example.

EDIT: I missed out walking bass lines! Ever since bass was invented in the late 15th century, it was a kind of not-very-melodic supporting role. But jazz musicians made it a much more active part of the ensemble, with melodies and slides and stuff, and this was carried over to an extent in popular music. It occurred to me just now listening to 'She's So Heavy' by the Beatles but that probably isn't the best example.

Song Structure

We're so used to instrumental solos that it's easy to forget that they must have been developed as a structural tool at some point somewhere. They were originally improvised or semi-improvised sections for jazz musicians in the band to temporarily take the spotlight, and sometimes each member would get a solo in turn. The most common one in pop music is the guitar solo, which usually comes in about half to two-thirds of the way through a song and helps keep it fresh and lasts for eight bars. There's one in the Beatles' 'I Call Your Name', 'Shakin' All Over' by Johnny Kidd and the Pirates, and a piano solo in Gerry and the Pacemakers' 'How Do You Do It?'

It's easy to be reasonably precise about where they come in because early pop songs had a pretty rigid structure of verses and choruses. I won't lay it out here, but again it comes from an adaptation of the blues.

A final little jazz trick I'll mention is call and response, which refers to when the singer sings a short snippet of melody and the band 'responds' to it, sometimes with backing singers as well. It's really clear in 'My Generation' by the Who, when Daltrey sings something like 'People try to put us d-d-down' and the rest, along with the instruments, sing the response 'Talkin' 'bout my generation', and crops up all over the place. It's probably the single most obvious thing taken from jazz, because the exact same technique features in hundreds and hundreds of jazz songs.

It's not the case that all of pop's influences came from jazz, but they definitely played a huge part. From the late '60s, music really started to branch out, and we got progressive rock (which looks for influence in classical music) as well as more interesting folk-rock and other folk-influenced bands. But hopefully I've given you a brief impression of a few ways in which jazz influenced the development of pop.

Anyway I've probably missed out a huge number of other things, so if you want me to expand on anything just ask. For more information, check out Popular Music and Society by Brian Longhurst and Understanding Popular Music Culture by Roy Shuker.

The simple answer here is the usage of the "Blues Scale". All western music is generally influenced by either jazz or folk music (with both of those coming classical music) and sometimes the music is influenced by both. If you go through time starting from the 1950s, you will find some usage of the Blues scale and other influences of jazz in the works of Elvis, Jimi Hendrix, The Grateful Dead, Metallica, The Clash, Bob Marley, A Tribe Called Quest, and Daft Punk, among many other artists (but these are just a few rather famous artists that span very different genres of music). So I'd say, in my opinion, jazz music is the most important influence of all possible other influences to modern music. If you'd like to know more I'd be happy to help.

Here's a great book I read on the history of popular music: http://www.amazon.com/American-Popular-Music-Larry-Starr/dp/0199859116

I really recommend it.

I'd like to write a TL;DR about the period in question, but I'm afraid I'm no actual historian, nor do I know how to properly write about history.