Making international phone calls is easier than you might think. Once you understand the extra numbers you'll see, you can start calling Europe or, when in Europe call country to country.
Anatomy of a European Phone Number - Breaking the Telephone Code
First, you will need to know what the sections of a phone number mean. Let's say you want to make reservations at Florence's famed Uffizi Gallery. You will see the number on their website is written as 39-055-294-883, or sometimes you'll see it written as (++39) 055 294883.
(The single or double + reminds you to add your International Access Code, which for North America—the US and Canada—is 011.)
Breaking down the phone number for the Uffizi, 39 is the country code for Italy. 055 is the city or area code for Florence (Firenze). Country Codes can vary from 2 to three digits. City codes in Italy can vary from 2 to 4 digits. The rest is the local telephone number, which also may vary in the number of digits.
Making the International Phone Call
No matter how you are calling—by cell, landline, Skype or from a European number, in this case outside of Italy, it's an International call.
Here's how to make the call:
You must add the International Access Code. For the US and Canada, this code is 011.
So to call the Uffizi and request tickets from the US, you would dial:
011 39 055 294883
in other words:
(Access Code) (Country Code) (Area or City Code) (Number)
Some countries don't use an area or city code, in which case you can omit this number.
If you were within Italy using a phone with an Italian SIM card, you'd simply dial the number: 055 294883.
Dialing North America from Europe
Simple. To call home, just dial 001, then the American number (area code, then a local number). The 00 is the direct dialing prefix, and the 1 is the country code for North America (Canada and the US).
Calling from Country to Country in Europe
For making calls in Europe from country to country you can use your U.S. mobile phone with roaming, which is often expensive so check with your carrier. You can buy a cheap cell phone in Europe with a local SIM, or, if you have an unlocked cell phone and are planning multiple country vacations, you can get a sim card in most countries from a store or kiosk.
If you just make local calls and receive email, a SIM card with 20 or 30 euro credit will probably do.
In some countries, you can drop the area code if dialing landline to landline within the same area code. In other countries, you have to dial the full number. To reduce the confusion you can just dial the full phone number, including the area code or prefix as a general rule.
Problem Solving Calling Abroad
The website, How to Call Abroad, is a good resource for problem-solving. Issues such a call consistently not going through, looking up any country's calling code, and checking which countries us the North American Numbering Plan can be found on the site.