Impedance in headphones vs. speakers - confused

Bombaywall, thanks so much. I really appreciate it. I admit I had to read your post a couple of times to finally understand everything but I think I got it. Math was never my strong point.

So my new integrated amplifier states in the manual that only headphones with a minimum of 50 ohms should plugged into the headphone jack. So I take it this means that there is not a lot of current available through the amp powering the headphones? But an acceptable power supply?

The one thing that I find confusing though is that most home amplifiers that can handle lots of current are usually expensive. My guess would be because of dealing with the heat generated. But in terms of headphones, an iPhone can easily muster enough current to power a lower impedance headphone such as 16 or 32 ohms. Is it because the amount of current is so small at that point thats not really a big deal? But voltage is because its battery powered?

Hope I got that all right. :) I'm still learning.

Ok. This is all starting to make sense to me now and I really appreciate your very clear and helpful answers. This has been bothering me for quite a while so it feels good to start to understand it :) Thanks!

Theres's only one thing that I still don't understand. Why is it that an iPod or iPhone won't get stressed out delivering the current required for 32 ohm headphones (apple earbud's or a pair of Grado's for example) but my $3000 integrated amplifier which requires a minimum of 50 ohms has a problem with it? Yet the integrated amp can apparently supply enough current to power 4 ohm home speakers with no problem.

I really appreciate all the help! Thanks everyone!