I always hear people talk about how great The Beatles were, but I've never really taken the time to listen to much of what they have to offer. Any suggestions?
They are the 1960 English rock band right? They have so many classics. My favorite is the song called "Yesterday".
When did they become so huge?
They became very popular in 1963. They called it "Beatlemania"! Lennon and McCartney were very important in their popularity.
Oh wow, very interesting! What genre of music do they make?
They have many styles, like rock, Indian and even psychedelia. It probably contributed to their broad success. 
I would imagine. Being able to put out quality content in that many genres is fascinating. Are they still around at all?
No, the original tri of Lennon McCartney and Harrison went through a bunch of drummers, but many of the members have passed away now.