Many handheld Air-band radios/scanners are available today costing from just a few pounds to several hundred pounds. The cheaper air-band radio is usually able to pick up transmissions from civilian aircraft but are limited by the absence of accurate tuning so you have little idea of what frequency you are listening to and you are often tuned into more than one frequency at the same time.
A scanner, while more costly (about £70 and more), usually consists of a keypad to enter a particular frequency, LCD display, a memory which will remember stored frequencies, and the ability to scan through a certain set of frequencies. There isn't any frequency overlap that you get on the cheaper Air-band radio. You can get either a hand-held scanner, ideal for air shows etc, or a larger desk top/base station model.
The cheaper scanners can receive just the VHF aircraft band (108-137MHz) which is ideal if you want to listen to civilian aircraft transmissions. More expensive models allow you to receive the UHF aircraft band (225-400MHz) in addition to the VHF aircraft band. These scanners can receive military aircraft transmissions as well as civilian. In order to listen to the Red Arrows leader at an air show you would require the UHF band.
Both VHF and UHF are short range, line-of-sight radio transmissions which are restricted to a maximum range of about 200 miles due to the curvature of the earth and usually considerably less depending on the surroundings (hills etc). Some scanners can receive the HF (short wave) aircraft band in addition to the VHF and UHF aircraft bands. The HF band is not restricted to line-of-sight radio transmissions and can be used to monitor Shanwick and the trans-Atlantic routes etc. There are also multi-band scanners which cover much more than just the aircraft bands.