No color

are you sure you're using component? (red/green/blue + red/white connectors) ?
if so, then it doesn't like a faulty cable or faulty component connection.

if you're actually using composite (red/white/yellow) then that is not supported.

Welcome to the  Discussion Forums.

To add further to what Chenks has advised you, a faulty component cable is unlikely to give you a B & W picture, so it sounds very much as though you may be trying to use composite cables. Although technically it's not the composite cables that are a problem but rather that you may be trying to connect to a composite device (yellow for video)

for clarity this is component

!http://nukoda.com/wp-content/uploads/2009/10/component-video-cable-5RCA.jpg!

and this is composite

!http://www1.istockphoto.com/file thumbview_approve/3189440/2/istockphoto_3189440_audio_video_cableisolation.jpg!

I've got my hands on my dad's Apple TV, but unlike my dad, the main TV in the flat I'm moving into is standard (not high) definition. Although it's 4:3, it is capable of displaying 16:9 images by using a letterbox. Would using a device like this with the appropriate cables be likely to allow me to use the ATV?

Regards

Probably, assuming it has either a HDMI connection (probably won't) or a COMPONENT connection.