Many people ask me why anyone would have been loyal to the king. After all, he was a tyrant wasn't he? That may be what we are taught now, but not everyone felt that way at the time. Anyway, if I had lived back then, I certainly can't be sure I'd wan't to replace a perfectly stable and successful government with a completely untested one. And many people who had been in favor of the revolution weren't at all happy with their new government once it was all over.
At the beginning of the American Revolution, separation from Mother England wasn't all that popular an idea. Only about one third of the population of the colonies was in favor of war, one third was opposed to the revolution and the remaining third was undecided.
The Loyalists were those people who didn't think their situation was so bad and remained loyal to the king. Almost all of the people living in the British colonies looked upon themselves as British subjects and therefor under the rule of King George III. It's only natural that those people would want to keep their national identity rather than throw it away in favor of the unknown, especally when they like the way their life is going.
But what about those horrible taxes? The taxes imposed by the British government on the colonists were certainly not enough to break anyone. The people were outraged simply because they hadn't had to pay taxes before then. The tougher taxes (which were nothing compared to what they had to pay AFTER the revolution) were even repealed after the colonists complained.
The reason taxes were imposed in the first place made them seem all the more fair to many. The taxes were intended to pay for the protection of the colonists from Indians and the French who objected to the colonization of land they considered their's (we don't need to get into issues of true ownership right now).
What about taxation without representaton? Each colony had a governer who was a go between for the people and the British parliment. The governer could go (or send someone) to speak to parliment on the colonists behalf. The British government felt that was ample representation and many colonists agreed.
Basically, the colonists in favor of revolution wanted to be treated like regular British subjects with regards to parlimentary representation, while not paying the taxes that other Britains did and while enjoying the greater personal freedoms they had living an ocean away from law makers. Of course there's more to it than that but you'd have to take things on a more individual basis. The Loyalists, on the other hand, were often colonists who were born in Britain (or had close family members who were born there) and appreciated the opportunities for land and jobs here while still feeling close ties to the mother country. Loyalists simply felt that there was no need for a change in leadership and, in fact, early on in the revolution many "rebels" were in favor of retaining ties with Britain while gaining a larger degree of home rule.
Emotions at the time were very strong on the subject of independance to the point where many Loyalists had their homes taken or destroyed (and sometimes they were even killed) by angry mobs. This is what forced many loyalists out of their complacency and into the army to fight against the rebel colonists.
Loyalists are often represented today as being traitors to their own people who tried to sell them out to a tyrant king. That just wasn't the case. The Loyalists were simply loyal to their government and didn't feel one third of the population should be able to decide for the rest simply because they were more vocal or more angry. But of course, the winning side writes the history.