“Snatchers ...They’re everywhere—gangs trying to earn gold by rounding up Muggle-borns and blood traitors, there’s a reward from the Ministry for everyone captured."
-- Ron Weasley (DH19)
The Snatchers were roaming mercenary groups of wizards during the second rise of Voldemort (circa 1997-1998) who earned gold from the Ministry of Magic (or Voldemort) for rounding up Muggle-borns and blood traitors (DH19).
When Ron was alone and on the run from Death Eaters, he was caught by a group of five Snatchers. He lied about being Stan Shunpike and as an argument ensued. Ron was able to grab a wand and Disapparate, eventually using the deluminator finding Harry and Hermione in the Forest of Dean (DH19).
After Harry uttered the "Taboo" name of Voldemort inside their tent (DH22), Harry, Ron and Hermione were found by a group of about six Snatchers including werewolf Fenrir Greyback and a wizard named Scabior (DH23). They had already captured Dean Thomas and a goblin named Griphook, and killed Ted Tonks, Dirk Cresswell, and a goblin named Gornuk (DH15, DH22).
Hermione had disfigured Harry's face with a jinx, and he told the Snatchers he was a Slytherin named "Vernon Dudley." But Scabior found some glasses in their tent and Fenrir suspected his real identity. Instead of taking the prisoners to the Ministry for gold, they went to Malfoy Manor where they hoped Voldemort would reward them. But the Dark Lord wasn't there and Draco refused to identify Harry. But Narcissa recognized Hermione and Ron, and Draco confirmed their identities.
Bellatrix Lestrange saw that the Snatchers holding what looked like the Sword of Gryffindor, which was supposed to be in her vault at Gringotts. When the men wouldn't relinquish the sword, Bellatrix stupefied the Snatchers and ordered Draco to drag them outside the Manor house. She didn't stun Greyback, but drove him to this knees and eventually released him. Bellatrix tortured Hermione with the Cruciatus Curse, and threatened to give her to the werewolf. But eventually Dobby arrived, summoned by Harry's magical mirror, and the hostages including Luna Lovegood and Mr. Ollivander were rescued.
Hermione had wisely stuffed her beaded bag containing all their belongings into her sock before the Snatchers found it.
Commentary
Etymology
early 13c., "trap or snare" or "make a sudden snap or bite" (at something), of uncertain origin; perhaps from Middle Dutch snacken "to snatch"