They are referring to the Dark Mark tattoed on Draco's arm.
Harry believes that Draco is a fully-fledged Death Eater, who showed Borgin the Dark Mark to intimidate him and have him take care of the Vanishing Cabinet.
However, what they both saw was Draco showing off his arm. From Harry's perspective, he thinks it is obvious that Draco showed Borgin the Dark Mark. Hermione, on the other hand, believes Draco is too young to be a Death Eater.
The book makes things a bit more clear.
“But what about when he said, ‘Don’t forget to keep that one safe’?” asked Harry for the umpteenth time. “That sounded to me like Borgin’s got another one of the broken objects, and Malfoy wants both. Malfoy’s father’s in Azkaban. Don’t you think Malfoy’d like revenge? He’s a Death Eater,” said Harry slowly. “He’s replaced his father as a Death Eater!”
There was a silence; then Ron erupted in laughter. “Malfoy? He’s sixteen, Harry! You think You-Know-Who would let Malfoy join?”
“It seems very unlikely, Harry,” said Hermione in a repressive sort of voice. “What makes you think … ?”
“In Madam Malkin’s. She didn’t touch him, but he yelled and jerked his arm away from her when she went to roll up his sleeve. It was his left arm. He’s been branded with the Dark Mark. He showed Borgin something we couldn’t see,” Harry pressed on stubbornly. “Something that seriously scared Borgin. It was the Mark, I know it… he was showing Borgin who he was dealing with, you saw how seriously Borgin took him!”