There were numerous such songs in 50's. All of them are characterized by the same rhythm and typically they are very similar. Is there some name for this kind of songs?
I am not sure that there is a specific name for this style that you could google and get a comprehensive list. It is very common for the time. One of the first things I did was to look up the chord progression. It is a very standard progression for the time 1 6 4 5. This wiki article is a good place to start looking for similar songs: 50s Progression
It's simply a song from an American rock and roll album from the year 1960. There is no more specific "genre" or term for this kind of song. Although there is a technical musical term for the Everly Brother's singing style and their means of arranging melody and harmony. It's called close harmony, but it is not particular to the Everly Brothers and 1960. There are examples of close harmony singing all through the history of American and British pop music.
Look for songs recorded near this time that were written by Boudleaux and Felice Bryant, the famous husband-and-wife songwriting duo. Look for albums and songs from this time period, from artists and acts that are white (Caucasian), and recorded in the Southeastern USA: in Nashville, Muscle Shoals, Memphis or Atlanta. Don't look for songs recorded in Los Angeles or New York or Chicago. They all had very different styles at this time.
AllMusic.com lists similar albums to this one at this link:
This list cites albums by Eddie Cochran, Gene Vincent, Buddy Holly, Jerry Lee Lewis, Elvis, Ricky Nelson.
Also see the Louvin Brothers. They were an older act that was more country rather than rock, but they employed the same kind of close harmony that the Everly Brothers used.
On Wikipedia, the Everlys are characterized variously as doing rock, pop-rock, country rock and rockabilly music. This song in particular shows a strong blues influence, which was common in early rock of the 50's and 60's (understandable since rock and roll developed directly out of the blues). It also features the country-music derived close harmonies that were the Everlys trademark. In general, this song has a very typical sound for the pop-oriented rock of that era.