The Matter Paddle was a device that allowed instantaneous transport over interstellar distances.
The paddle proved capable of converting several individuals into digital information and then transmitting them as light beams to another point in space. It was essentially a matter transporter.
Kryten's first journeys were within Red Dwarf, showing the crew how the matter paddle worked. He then took Lister on a short journey to Starbug, but made a small error that landed them in the shower instead. Although Rimmer was a hologram, he was able to touch the paddle using his light bee.
The paddle was able to home in on S3 planets within 500,000 light years, which led the crew to visit Waxworld, a colossal Waxdroid theme park. When Kryten and Rimmer arrived- travelling on ahead to determine if the planet's atmosphere would be safe for the organic crewmembers- they sent the paddle back for Lister and The Cat to join them. The paddle was then confiscated by the occupants of Villainworld. Caligula tried to force Lister to operate the machine for him, which allowed Lister and Cat to escape.
When Rimmer sent the heroes of Waxworld against the villains, Kryten was able to steal back the matter paddle. The crew were then able to return to Red Dwarf. ("Meltdown")
It was later suggested by Noddy in "Back to Earth" that the Boys from the Dwarf could use the paddle for transport, to which the by then frustrated crew shouted "Back on Red Dwarf!" ("Back to Earth, Part Two")
Behind the scenes
The matter paddle prop was created using four hairdryers.
With a range of 500,000 light years the matter paddle could transport the user anywhere in the Milky Way galaxy but did not have the range to reach the nearest neighboring galaxy Andromeda which is approximately two million light years away and closing. Consequently with the correct coordinates the matter paddle could return the Boys from the Dwarf to Earth.