https://en.wikipedia.org/wiki/Amateur_radio_international_operation
(For Americans and Canadians: Operate in each others countries. You're good-to-go.)
Just check out that page, and if you still have any doubts, go to these other links below, but be warned that they can only add to the confusion:
European Conference of Postal & Telecommunications Administrations (CEPT)
FCC Notice regarding CEPT: http://www.arrl.org/files/file/cept-ral.pdf
In the Americas, Canada, The United States, and Peru are all CEPT signatories.
When operating in CEPT countries, American Hams must carry:
- Original US license
- Proof of US citizenship
- A copy of the FCC public notice when operating in CEPT countries.
International Amateur Radio Permit (IARP)
Used between signatories of Organization of American States (OAS) agreement A-62.
Not needed if both nations are signatories of CEPT, or have a reciprocal operating agreement, which takes precedence.
Good for 365 days.
ARRL pdf to obtain an IARP:
http://www.arrl.org/files/file/VEs/IARP%20Application%20March%202015%20interactive.pdf
It's not needed by Canadians operating in the US or Americans operating in Canada, due to the Canada-US Reciprocal Operating Agreement of 1952, and oh yeah, CEPT.
ARRL page that's a good place to start regarding operating on other countries:
http://www.arrl.org/international-operating
RAC.CA FAQ on CEPT/IARP:
http://wp.rac.ca/operating/cept-and-iarp-faq
An important point is the final answer on the page, explaining why Canadians in the US and Americans in Canada don't need an IARP.
It explains that the Canada-US Reciprocal Operating Agreement bypasses that requirement.
Of course, even without the 1952 reciprocal agreement, both US and Canada are CEPT signatories, so that would apply even if there was no reciprocal agreement.
The agreement: http://wp.rac.ca/operating/canada-united-states-reciprocal-operating-agreement
It should be noted that although there is also a reciprocal operating agreement between the US and Canada, American operators in Canada may still need to carry the same documents required by CEPT. This is where things get fuzzy as there are both pages that say Americans and Canadians Do or Don't need to carry a CEPT document. -shrug- Just print it out and carry it to be safe.
Also when in doubt, find out what the authoritative organization(s) in the destination country say, not what is being said in your own country.
But wait, there's even more information in case you haven't drowned yet!
ARRL has links to other countries' information about their international agreements. Note that some of the links are broken, but if you're traveling abroad, check this page.