This is an important Hungarian tradition left over from the 1848 Revolution against the Austrians. Because the Austrians celebrated the defeat of the Hungarians by toasting and clinking mugs of beer, the Hungarians avoid this particular behavior and actually banned it.
When toasting in Hungary, begin by making eye contact, raise your glass to eye level, say “Egészségedre” („cheers“), and take a drink. But be very careful: mispronounce “egészségedre” and you might be saying 'to your arse' instead of 'to your health'. Always ask a Hungarian for advice on pronunciation!
Further research indicates that recently it was claimed that the ban on clinking was meant to last only 150 years (ending in 1998). The Hungarian media picked up on this version of events, but many Hungarians still adhere to the rule. Clinking with any other alcoholic beverage, however, such as wine, champagne or hard liquor is fine.