I have a US Dell Inspiron Mini 10 and I'm travelling to a European country that has 220 v power. How do I find out if I just need the adapter plug or if I need to have a power converter from 220 to 110 as well? The 3 prong plug that came with the mini says Input 100-240v so I'm assuming I can just plug it into the adapter for that country (even if there are only 2 prongs)?
You said it.
Modern switched mode power supplies are generally capable of coping with any voltage from around 100 to 240 volts. Older PSUs built in to computers often had a 240/110 volt switch (many a time I've had to replace one because the owner switched it to 110v in the UK and blew it up...) but those have been dropped in favour of auto-switching.
As long as the label on the power supply looks something like this:
(note the INPUT line - 100-240V ~ 1.5A(1,5A) 50-60Hz) then you should be perfectly fine to use this power adapter in any country. If you are thinking of traveling to the UK frequently it may be worth investing in a UK power cable portion of the adapter (just the thick cable that goes between the wall and the power adapter) to save using socket adapters all the time.