ANR is "Active Noise Reduction". The headset is powered either from the aeroplane's electrical supply or by batteries and a computer within it plays "anti-noise" (effectively the inverse of the external noise) into the earphones which cancels out the external noise. They can be extremely good. I use Sennheiser and David Clark on a regular basis and can recommend either. However, they're unlikely to fall within your price range as they tend to be £400 plus.
"Passive" headsets are the traditional type which are quite simply a headset which also acts as a set of ear defenders, almost certainly what you're using at the moment. You should be able to find a decent one for around £200. Once again, I would recommend David Clark. You tend to have to pay a bit more but they will last a lot longer.