Saul grabs Samuel's robe, and accidentally tears it (15:27).
Image from HTML Bible
Chapter 15 is pretty straight forward. Basically, God (aka Yahweh) remembers that Amalekites did some pretty mean things to the Israelites a few hundred years ago, so He orders that they should be exterminated - elderly, men, women, children, infants, livestock, EVERYTHING).
Saul does a few things wrong however, he actually let a group of people called the Kenites go, because they were actually kind to the Israelites even though they were living with the Amalekites. I don't think this is what got Yahweh pissed off, though. What pissed Yahweh off was that Saul didn't kill everyone.
Saul had the gall to let the Amalekite king live (though all other Amalekites were slaughtered - men, women, children), and Saul decided to keep the best of the animals.
This pissed Yahweh off to no end, and He even said that He regretted making Saul a king. Imagine that - God has regrets!
Now, to be honest, Saul did this because the people asked. But, this is not what the Supreme Leader asked, via His spokesperson Samuel.
When Samuel learned that Saul had not followed the God's commands to the letter, He chewed out the king, in a display that showed who the real power was. The power isn't the king, who is a secular authority of sorts, but through the priesthood and especially via Samuel. In the picture above, Saul is shamed for his bad behavior and grabs the mantle of Samuel's cloak, accidentally tearing it.
After tearing the cloak, Samuel sees symbolism in it and says that God has torn the Kingdom of Israel away from you, Saul, and has given it to your neighbor.
The chapter ends in a final morbid scene. Saul, who has been stripped of his kingship, approaches King Agag of the Amalekites - their last surviving member. "Surely, the bitterness of death has passed," said Agag.
Saul replies, "As the sword has made women childless, so too will your mother (who is probably dead by now) be childless among women." With that, Saul cut Agag into many pieces.