which ethernet cable should I buy for connecting router to Apple TV?

A CAT6 cable would be suitable and affordable. You could use a lesser standard (CAT5) but the price difference is insignificant.

-Measure distance 'Router to AppleTV'; remember to keep Router & AppleTV unit 1-foot away from TV & any Sound Amplifier.

Cable should 'easily lay' between Router & AppleTV to prevent Tugging on the Connecting Clips'.

-Current Ethernet Standard is Cat-6.

-Avoid 'bundling 'any' cables together to avoid induced Signal interference or physical TUGGING.

-All Cables should be 'firmly Clip' connected, then 'loosely laid' to prevent Tugging on Clip Connection.

Tie Wraps or Tape should not be used to Snug 'any' cables together.

I buy Walmart Cables because they

are Packaged; still I examine Cable

Connections to be sure the Sheath Ends are Firmly attached.

Here's to reliable viewing!

ChipDale