In North America, the phone number of the form 1-areacode-number is interpreted as:
1 - North America (and some of the island territories, I believe)
areacode - your general geographic area
number - you phone number
Somebody calling you from overseas has to dial a code before the '1' to call you. With a mobile phone, they can simply dial a +. '+' is used to indicate that you must dial the international access code.
For example, somebody calling you from the UK would dial 00-1-areacode-number. From South Africa they would dial 09-1-areacode-number. You would publish your number as +1-areacode-number.
Generally, from a cellphone you can simply dial + before the country code.