The remote "remembers" whether or not it has turned a component on or off. It remembers the state of each component.
What
can confuse the remote is if you walk up to a component and turn it on manually. No remote can possibly know that you have turned that component on. There is only one-way communication between remotes and components. If you use the remote for everything you won't have any problems.
If something gets out of sync, pressing the help button will fix it. It asks you a few questions such as, "Is the TV on?".... "Now is the TV on?"... etc. and it will get things back in sync.
I literally have not touched any of my original remotes since I setup the Harmony, not
once. It's one button operation for everything. One of the more complicated setups I have is my Xbox. Due to the limited inputs on my receiver and the lack of upconversion, it's a multi-step process. When I hit "Play Xbox" on the LCD screen the following sequence of events occurrs: Receiver On > TV On > TV Video 1 > Receiver VCR 1 > Receiver CD > Receiver Digital Optical In 1. All with one button. It is very nice.