This depends a bit on where your going. Some places are very much cash cultures, and even though you might be able to use a debit card technically, very few places will accept it. For example, I spent two weeks in China last summer. Some large businesses (hotels, airlines, chain restaurants) would accept mu debit card, but a lot of smaller vendors, stores, and restaurants just worked on a cash basis.
When I went to England five years ago, I had very much the opposite experience. It was like being in the U.S. I carried a little cash on me for small purchases (like food from a street vendor), but I used a credit card for anything substantial.
When you swipe your card, your bank will convert the amount based on the current conversion rate between USD and the target currency. They'll also typically charge you a fee (1%, maybe 3%) for the currency exchange. Your bank should be able to tell you what that fee is.
It works the same way with ATMs. You can withdraw money in the local currency using your American debit card.