Commodore Infra Red Joypad.

This Joypad was packaged with every CDTV unit sold and was the standard way to control the CDTV.
It is powered by two AA (RX6) batteries and contains a full numeric pad, two mouse/fire buttons and four arrow keys for control as well as CD/TV, Genlock, Power and Mouse/Joystick buttons.

What does the CD/TV, Genlock, Power and Mouse/Joystick buttons do?

CD/TV: This button is for use on CDTV`s with an Ariel in and RF TV out, Pressing the CD/TV button will change between the CDTV and normal TV ariel.
Genlock: If you have the optional CD1300 or CD1301 Genlock unit installed this button is used to change between the Genlock modes "CDTV Only" "Mixed" and "External Source".
Power: This button turns the CDTV unit on and off, The remote does not require this as it uses power only when a button is pressed.
Mouse/Joystick: There are two main ways to control software, Either by using a mouse or a Joystick, This button lets you change the joypad into a mouse if needed. If your remote is not working with a program you should press this button in case the program is expecting a mouse.

If you loose your joypad then there are a few companies selling them, They include:
Eyetech: Its the only CDTV item they sell.
Vesalia: They sell Joypads (Sometimes Josnet) and more CDTV items.
And
Old Software.com: Stockists of lots of CDTV items, E-Mail for joypad info.