A "45" is a small vinyl record - usually 7 inches in diameter. It typically contains one song on each side - one on the front ("A side") and one on the back ("B side"), which are about 7 minutes long. There are also 12-inch 45s, which were mainly produced in the 1970s and 1980s.
45s were developed post-WWII. They were made of vinyl, which was much less fragile than earlier formats. 45s were made to be played in jukeboxes - both in establishments and at home. Because they were so portable, durable and inexpensive, 45s changed the face of music consumption. They remained a standard format until the 1980s.