Clearly her previous iPhone experience was paramount.
But obviously, it wasn't like that when he first used it. It took time. For example, it took forever to teach him that if the hand that is holding it is touching the screen a little bit, then it won't properly register new touch events from the other hand. Eventually he just learned this, but it took a long time.
The other thing to note is that children actually learn how to use computers far more rapidly than you might think. It's not just devices like this, where people will assume that it is the multi-touch and intuitive interface that makes it easy for them to learn it. This helps, sure, but I was shocked when, after a mere two hours or so of usage, my two-year-old had learned how to navigate Flash games in a browser using the mouse. I was about to restart the game he was playing by clicking through the menus but no, he grabbed the mouse and did it himself, to my amazement.
Bottom-line: although I sense that many people will assume it is Apple's superior design that is responsible for this child's proficiency, I think that the combination of prior experience, plus the very quick learning abilities of young children, is a more important factor.
An actual computer? Check out the games on pbskids.org - esp. the Curious George ones. For under <3s, you will probably need to "drive" the mouse while the kids tell you what to do.
The iPhone has a myriad of kid-friendly apps suitable for the toddler set. Try AnimalSNAP, Scribble, PreSchool Playground, Matches, and Wild Animals jigsaw. While obviously the iPhone/iPod Touch/iPad is not a substitute for real play, it has certain advantages, such as encouraging seriously fine motor control, a level of abstraction, and being a compact source of amusement for plane rides.
Also check out the kids books "with synchronised images" from audible.com. It allows the iPod to pretend to read books to your child. Again, not a substitute for you doing this, but a good tool for your emergency arsenal.
I have to say though, that the most educational thing on the internet for the preschooler is youtube. There is nothing like, say, talking to your kid about the instruments of the orchestra and then being able to pull up somebody playing the tuba right there and then. Indispensable.
That said, get your timing right. I kept my kid away from the TV till she was two, and away from gadgets till she was three. I think the level of engagement you get by introducing things at the right time is much higher than if you expose too early.
Open up notepad. Maximize the window. Change the font to 72. Turn on caps lock. Let them have free reign on the keyboard and tell them what letter they are holding down. After about two months of this they are pretty proficient at the keyboard and at their letters.
For mouse skills fullscreen Microsoft Paint. Same results.
Secondly, there's probably all sorts of great educational stuff for them to check out but all I've really done is put him on some simple Flash games. Examples:
http://www.kongregate.com/games/Teagames/tg-motocross-2
http://www.kongregate.com/games/light_bringer777/learn-to-fl...
I always tell people I learned how to navigate MS-DOS when I was five from watching my cousin do it. I wanted to get on the computer and play games, and I just remembered what he’d typed. I actually have no idea how true this memory is. But I wonder if CLI can feel just as natural to young children.
But if her previous experience had been with Windows 7, would she be similarly proficient on an HP Slate?