Guillotine execution is the form of Persona Fusion within Persona 5, where it takes two Personas and creates another of a differing Arcana. After your first few visits to the Velvet Room as part of the story, you'll earn access to it through dungeons and out in the city. Entering the Velvet Room in either of these settings will allow you to talk to Igor and utilize his abilities to create new Persona, which may be necessary for some Confidant missions that you didn't earn a Persona for in a previous dungeon run.
Or, if you're going for the Strength Arcana, Justine/Caroline will require you to return with Personas equipped with specific skills (that will require fusion or manipulation of their skill sets with cards).