The following countries
drive on the left
and (in theory
on the right
Therefore, all other countries drive on the right, and (in theory) pass on the left.1 This includes all of the Western hemisphere (except parts of the Caribbean), the Middle East, Continental Europe, the former USSR, China and Northern Africa. Former British colonies generally drive on the left (with the exception of Canada and the USA), and former French, Spanish and American colonies generally drive on the right. If you total it up, about 2 billion people drive on the left (or would, if they drove), and 4 billion on the right.
This all raises the question of why the discrepancy? Evidence2 exists to indicate that the Ancient Romans drove on the left. There is also reason to believe that it was more natural for (commonly) right-handed people to ride horses on the left. First, as Withnail indicates, so they can mount at the side, and also so that their right (sword) arm is free to take swipes at oncoming enemies. So then why would anyone drive on the right? Most people trace the answer back to Napoleon, who decreed (among many other things) that in his empire, everyone would keep to the right. Some claim that this was simply to be revolutionary, although it is likely that France had always been like that, ever since Roman times. The reason being that wagon teams with postilion riders were common in pre-revolutionary France, and it was more natural for these to drive on the right, as the postilion rider would sit on the left side (to be better able to control the team with the whip in his right hand) which would afford a better view of the road if the wagon were on the right.
As Napoleon's armies conquered more and more of continental Europe, this rule took effect over this same area. Even after his Empire vanished, the right-hand rule stayed on.
Other parts of the world largely adopted either the right-hand or left-hand rule as they were colonized by various European nations. While this wouldn't at first seem to explain the right-side preference of North America, realize that much of North America was colonized by the French before the British. They controlled a huge swath of land from Québec all the way down to Louisiana; much of this was never under left-hand rule, and it ended up that the remainder of the US and Canada followed this lead, rather than the other way around. This was probably helped by the fact that postilion-type wagons were also common in early North America.
Trains in the Channel Tunnel run on the left.
1 There is the interesting case of Tajikistan, where driving is officially on the right, but in practice can vary from day to day and from region to region. Advice for the tourist is to watch out for oncoming traffic, and make sure that which ever side of the road they are on, you are not.
2 Two pieces in particular; a type of Roman coin shows chariots passing each other on the left, and the remains of a Roman quarry near Swindon, England that has ruts in the ground which are deeper on the right side of the road as you head towards the quarry, indicating that heavier loads were carried (ie: away from the site) on that side. Hence the left.