The Beatles changed pop music.

Rolling Stone said:

"The impact of the Beatles – not only on rock & roll but on all of Western culture – is simply incalculable … [A]s personalities, they defined and incarnated '60s style: smart, idealistic, playful, irreverent, eclectic. Although many of their sales and attendance records have since been surpassed, no group has so radically transformed the sound and significance of rock & roll. ... [they] proved that rock & roll could embrace a limitless variety of harmonies, structures, and sounds; virtually every rock experiment has some precedent on Beatles record"

The Beatles influence came from their massive popularity. They were the most commercially succesful act of the 60's. The band had to stop performing live because they couldn't find venues large enough and the crowds were too loud. Either the Beatles or Michael Jackson have had the most Top 20 songs on the Billboard charts, and certainly the Beatles had that record first. Their songs are still being covered and the surviving members still have successful careers despite a lack of recently successful albums.

Another reason the Beatles are influential is because of their large catalog of songs. The band wrote most of its own music, which was unusual at the time for many commercially successful acts, and wrote cross a large varieties of sub-genres. Initially the band released simple, catchy pop-rock tunes, and then over the course of their career developed or popularized multiple different sounds. The Beatles gained popularity with songs like " I want to hold your hand" and "She loves you." Playing Ed Sullivan show, which had an enormous reach, gained the band overnight popularity in America. They are widely acknowledged to have kicked off the "british invasion" which paved the way for bands that would become mega-stars like the Rolling Stones and Led Zeppelin. The Beatles sound changed as well, whether they influenced the change of American culture in the 60's or just rode that wave, they still were the most popular band at the time to be playing psychedelic music, which popularized the genre and got it airplay on pop stations. Even the Beatles album covers are seen as influential. The Sgt. Peppers album cover has been recreated by a number of artists and bands. The Abbey Road cover, with the band walking across the street, is iconic.

In short, the Beatles are influential because they were popular. People listened to them and expanded on or copied what they were doing.

I'm sure others can go into more detail on any of these points and Beatles trivia, of which there is a ton, but that can get you started.

edit: forgot a sentence

You asked about "socio-economic" influence, so I thought I'd share two fun examples of how the Beatles' commercial success had major effects beyond music and popular culture.

1. Their touring activities and overseas record sales in the mid-1960s generated so much foreign exchange for the UK that it is thought to have staved off a UK currency crisis. This unexpected flood of dollars gave Harold Wilson's new Labour government time to put in place measures like the first UK Race Relations Act and the UK National [economic] Plan (both in 1965), the building of lots of high-rise public housing, and various other policies. The Beatles' last commercial tour was followed by a 15% devaluation in the pound in 1967 which made much of the Labour government's economic policy much more difficult.

Source: Financial “deglobalization”?: capital flows, banks, and the Beatles - speech by Kristin Forbes, Bank of England (2014)

2. The large profits the Beatles made for their record label, Parlophone, played an indirect role in the invention of the CT scanner. Parlophone was owned by EMI, who in those days were a conglomerate. One of things they used their cashflow from operations to fund was their central R&D lab, where between 1967 and 1971 Godfrey Hounsfield developed the first commercial CT scanner. Much of the R&D was funded by the UK government, but some was funded by EMI; EMI also spent a lot of money manufacturing and marketing the product. In the end EMI failed to make a commercial success of the scanner, but GE and Siemens did, building on the groundbreaking work of Hounsfield (who got a Nobel Prize and knighthood for his invention).

Source: "Do we really need to thank the Beatles for the financing of the development of the computed tomography scanner?", Vos P & Maizlin Z, Journal of Computer Assisted Tomography, 2012

"EMI and the first CT scanner", Alexander, R and Gunderman, R, J of American Coll of Radiology 2010

One thing about the Beatles that isn't immediately clear to people in 2016 is just how popular and important to youth culture pop music was in the 1960s, and how much of a fuss youth culture caused in the 1960s. Remember that, in 1964, when 'Beatlemania' exploded in America, there was no internet (I mean, Al Gore was only sixteen!). That means no internet memes or widely shared YouTube videos. Television networks largely thought in terms of broad appeal to all demographics at once, rather than the more niche programming you get today, so it was unsatisfying as a source of identity to youth in comparison to today. There were no home computers, let alone home computers with the processing power needed for No Man's Sky or whatever. Home videotaping wasn't a thing, and, in fact, television stations were only beginning to see their productions as being something beyond cleverly produced live theatre.

In contrast, the 7" vinyl single was reasonably cheap and easy to obtain, and a young person could collect them in a way that said something about who they were, and could play them over and over again in a way they couldn't for most other sources of pop culture, if their local radio stations weren't that interested in playing it. This meant that music was fairly close to the common cultural currency for the average teenager (and there were lots of average teenagers around in 1964, thanks to the rise in the average birth rate after World War II that led to the phrase 'Baby Boomer' - youth culture had never been such a big part of culture in general before this era).

And so in this 1960s music industry, where music is an important source of cultural identity for youth, the Beatles were the most popular musicians. And because the particular set of strengths the Beatles (and their producer George Martin) had were so popular and thus so widely imitated, they became influential long-term. As others have said, this sheer popularity led to some unlikely paths of influence for the Beatles - even if indirectly, they likely helped EMI fund the development of the CT Scanner, as /u/AbpStigand points out.

Musically, you could probably find some influence stemming from every single Beatles song - even 'Cry Baby Cry', one of the more forgotten tracks on their 1968 self-titled double album, is pretty clearly an influence on Jet's 'Look What You've Done' and Radiohead's 'Karma Police'. But I'd argue that there's four main ways in which the Beatles were particularly influential in putting together things we take for granted in music today and don't even think about.

Firstly, take our modern idea of 'the band' - a group of musicians with some sort of vaguely abstract name who may or may not have more than one singer, and who probably have a line up of one or two guitars, drums, and bass). There were examples of the band before the Beatles - the Crickets being the most obvious - but The Beatles absolutely popularised that idea (as I discuss elsewhere, including on the AskHistorians podcast and this comment thread here).

Secondly, before the Beatles, pop music structurally didn't really have choruses. The Beatles, musically, put together two sets of influences: a) rock'n'roll, in which there was a verse which varied and a chorus that stayed the same, where both verse and chorus were usually founded on the same chords - probably a 12-bar-blues form - and if there was variation from this it was an instrumental solo over the same chords (see 'Tutti Frutti' by Little Richard or 'Roll Over Beethoven' by Chuck Berry or 'Hound Dog' by Elvis); and b) Brill Building pop, in which the structure of the song was often based around AABA form, a la the Tin Pan Alley pop that's nowadays called the Great American Songbook and sung by Rod Stewart and Michael Buble. AABA form didn't have big stonking choruses (e.g., "She loves you, yeah yeah yeah" etc) but instead usually had a refrain, a single line sung at the end of every verse (e.g., the title line in Goffin/King's "Will You Still Love Me Tomorrow?" or Cole Porter's "Anything Goes" is a refrain). Because the Beatles (and, reasonably simultaneously, the Beach Boys) basically took the more complicated structure of AABA form, and added the choruses they loved from rock'n'roll, they solidified the modern pop song structure that is still used today with variations by the likes of Katy Perry or Taylor Swift or Bruno Mars.

Thirdly, the Beatles were at the vanguard of a profound change in popular music where a recording ceased to be seen as simply a record of a performance, but instead became an object of art in of itself. Frank Sinatra's 1950s records were often recorded on two microphones, with Sinatra simply singing close up to the microphones, with an orchestra behind him, recorded by the same microphones. These recordings are simply records of a (fantastic) performance. In contrast, The Beatles' 1966 'Tomorrow Never Knows' is a studio creation - they built the song up using the recording studio itself as an instrument, feeding sounds through the machinery so that they were backwards or half-speed, or creating loops of tape that would repeat. Thinking of the recording as an object of art in itself, today, goes without saying - nobody thinks that Katy Perry and a band playing behind her performed that song in one take, with no alterations. Everybody fundamentally assumes that 'California Gurls' or 'Roar' are studio creations, that it's carefully put together to create a confection that could never be recreated absolutely faithfully by musicians playing it live. These days, it's entirely possible that most of the people responsible for creating a hit have never actually met each other, and that it's all done through sending ProTools files over the internet.

Finally, The Beatles did a lot to popularise and create the myth of authenticity in rock and roll. Before the Beatles, it was unusual that a group of musicians sang their own songs - instead, a group of musicians put in a studio would be given songs written by professionals by a artist and repertoire man working for the record company who would try to find the right song. So, in some of the first studio sessions for the Beatles, they were asked by George Martin to record the song 'How Do You Do It?', which was later a hit for Gerry and the Pacemakers. The Beatles were vocal in the studio about their disdain for the song, convinced that they had written better songs themselves. George Martin as a producer had a hunch that while the Beatles might not yet know how to write great songs, they definitely knew the source of their appeal and what attitudes and feelings would work with their new 'guitar group' sound. So instead of making them release their version of 'How Do You Do It?', he worked with the Beatles to refine their songwriting and let them be themselves. Because of the popularity of the Beatles, some 'authenticity' began to be attached to bands that wrote their own songs - their music was seen as somehow more real, more convincing. After the Beatles became massively popular, the professional songwriting industry experienced a big decline - Carole King who co-wrote 'Will You Still Love Me Tomorrow?', for example, began to find it hard to get popular bands to sing her songs, because they were singing their own songs, and so she decided to record music herself. This trend of bands and musicians writing their own material has receded in straight pop music now, but this idea of authenticity is still all-important in hip-hop (see the recent 'beef' between Drake and Meek Mill over Drake not writing his own lyrics) and indie music, genres of music where musicians are expected to write their own material (to some extent).

You may be interested in some of the threads collected here - What made The Beatles so commercially successful?