It's one of the weirder rivalries in English football. The distance between the clubs is around 40 - 50 miles, Palace are a London team and Brighton are a South coast team, and although it's not a derby game in the tradition sense of the word, it is always the league game we would look to first in the fixture lists.
The origins of the rivalry started in the mid 70's when Terry Venables took over as Palace manager and Alan Mullery was Brighton manager. The two men already hated each other and the clubs were of a similar level and had similar expectations for the season. The league games that season were fierce and we were drawn together in the cup that season as well. Their was a war of words between the two managers and the rivalry continued after that. (This was all before my time, and I've grown up with the Palace / Brighton banter).
Every club has their rivals and teams that they hate, it's one of the things that make a football match. Personally I find the Brighton rivalry all a little pointless as we have been the better team (in a higher league), for many of the last few seasons, and I get more wound up by the cl**ns up the road, but Brighton is now a tradition for us, and as long as we keep on beating them long may it continue ))