The 1960s are fondly remembered as a particulary rich time in music history. While the birth of rock and roll came a decade previous, it's the 1960s that saw rock and roll become a cultural phenomenon unparalleled before or since. As a result, there are a plethora of 60's music hits that are still alive and well today, as recognizable as anything from modern acts like Radiohead, Jay-Z, The White Stripes, and everything in between.
Of course when thinking of 60's music hits, you should start with the most famous band of the 1960s, and really the most famous band of all time: The Beatles. They originally formed in the late 1950s, went through a couple of personnel changes, and eventually settled upon the famous quartet of John Lennon, Paul McCartney, George Harrison and Ringo Starr in 1962. From there they went to Hamburg, which was a far more forward-thinking city in terms of music than anything in the United Kingdom at the time, and became a European sensation. (They did play some gigs in their hometown of Liverpool as well, but it's the Hamburg time that is remembered as being most crucial in forming the Beatles' sound and legend).
Later that year they finally hit it big with their first chart-topper, "Love Me Do." It hit number seventeen in the United Kingdom, and among 60's music hits it's still considered one of the best. Ironically, though, their first big hit didn't feature all four Beatles; Starr was replaced on drums for that particular song by a studio drummer (who had the good fortune of receiving royalties for years). It was the last time all four Beatles wouldn't feature on a particular song.
A year later they found themselves in America at the spearhead of the British Invasion, and produced a string of 60's music hits with their albums Please Please Me, Meet the Beatles, and A Hard Day's Night. The third single off of Please Please Me, "From Me to You," was their first number one hit and really began the birth of Beatlemania. Lennon and McCartney had penned the song together and would later admit that, as with much of the band's early work, they were simply trying to copy the style of previous rock and roll acts like Buddy Holly and The Everyly Brothers. Lennon once went so far as to say that the words to the Beatles early songs were mostly meaningless.
After that early period, the Beatles evolved considerably over the years. Following the success of "From Me to You" and, even moreso, "I want to Hold Your Hand," Lennon and McCartney became more experimental both lyrically and musically. The album Help! was a big step forward in that regard, and that progress redoubled with Rubber Soul and Revolver, ultimately culminating with The White Album and Sgt. Pepper's Lonely Hearts Club Band later in the decade. Those last two are considered by many to be the gold standard for 60's music hits and rock and roll as a whole.