4-Koma (Yonkoma) manga is the common manga style where there are four panels (ki, sho, ten, ketsu).
A "Moe" 4-koma is the type featuring cute girls characters (moe meaning "to be attracted to"). They often don't follow the ki-sho-ten-ketsu pattern, they just have four panels. Another common feature is a serious lack of male characters. Examples include K-ON, Hidamari Sketch, Lucky Star, Nichijou, Kill Me Baby, Yuyushiki etc. many of which were made into anime.
How did these get so common?
Answer: Azumanga Daioh by Azuma Kiyohiko (1999 - 2002).
Most of the above manga are more or less copies of Azumanga Daioh - a group of schoolgirls not doing very much apart from talking with each other. In order to differentiate from Azumanga Daioh, there are small differences. K-ON gave the girls instruments, Lucky Star had an otaku girl as the main character, Hidamari Sketch had girls living in a dorm instead of their homes.
Why did Azumanga Daioh get so popular?
Because it had a group of attractive girls with wildly different personalities (a loli, a sporty one, an energetic one, a shy one, a lesbian etc. which all became manga staples) and its comedic value. Also of note is the fact that it finished cleanly after four volumes unlike K-ON or Lucky Star.
This manga seriously changed the industry in the early 2000s.