The Kindle and other e-readers use what's known as electronic-ink (or a similar method of producing images). There is a documented reduction in average reading speed when computers are used. They are easier to read than most computer screens. The iPad claims to have similar quality, but I've only seen poor tests done to suggest this. If the iPad has any of the screen glare of the iPhone, I doubt it performs as well as electronic readers.
I do the majority of my reading in bed on my side. If the book is large, I prop myself up with my hand. I prefer to have my head on a pillow and hold the book in one hand. The light-weight of my e-reader (kobo) allows me to do this. It also allows me to turn pages with one hand and a click of a button. You'd be surprised how much a lazy person can appreciate this.
I still read both, and I like real books because we live in a materialistic world and I like stuff (owning electronic things doesn't feel the same - likely because it doesn't communicate social status). I don't know what I read faster on. One disadvantage of e-readers is they need to be kept charged. They won't die on a full charge, but I forget sometimes. You can read with it plugged into a charger, but the charger operates on a usb (I am sure there are adapters for outlets). I'm not sure what the best e-reader is. I got the kobo as a gift so I didn't reflect on the options and choose it over others. I do like it, though.
Computers don't always harm your eyesight, but they can cause eyestrain, headaches, fatigue, and other issues. This will vary depending on your monitor quality. I need to get a new monitor as mine is terrible. I also have a program that reduces the lighting of my monitor relative to the time of day (so I get less glare at night in the dark).
I don't have issues sleeping but there are some people who suggest using electronic screens before bed causes sleep difficulties. I don't have sleep issues. I've fell asleep after consuming medically unhealthy amounts of caffeine. I'm going off on a tangent because I'm likely procrastinating.
I don't really feel like I'm using a computer screen when I use an electronic-reader. You can also download books, which is nice. I'm not sure which readers can do more in terms of text. For instance, I don't know if you could read a mathematical textbook and/or graphs (while I'm certain the iPad can).
You should be able to view an e-reader in the store. If you read a chapter of a book (almost all of them have a few default books), you can see if it's for you. If computer screens converted over to electronic-ink formats (which probably isn't likely anytime soon, as there are better technologies I suspect, just cost issues), I'd be quite happy. There are studies on the relationship between screen-technology and the viewer, and some people believe it decreases the value of the experience. So maybe all our video games will be 20% better when screens improve.