The Gizmometer is an invention created by Professor Hubert J. Farnsworth and its main purpose is to scan people's brains. Farnsworth originally tries to use a guillotine to 'scan' Leela's brain, but Leela avoids it. When she asks for a way to scan her brain without removing her head, Farnsworth brings out the Gizmometer.
Note: This particular invention of Farnsworth's appears twice in the episode, but both times in Leela's dream. So it actually may not exist.