Radio waves ignore borders, so the ITU (International Telecommunication Union) coordinates how spectrum is used worldwide. Every country then licenses its own amateurs under its own laws. If you travel with a radio, the host country's rules apply to you, and they may differ from the ones you are used to at home.
ITU regions
The ITU divides the world into three regions, each with its own table of frequency allocations. The Americas, including the US, are Region 2.
ITU allocations. Countries often adjust them, so check the host country.
Most amateur bands exist worldwide, but not always in the same width. On 2 m, for example, Region 1 amateurs have only the lower half of what the Americas have. Band plans, power limits and permitted modes also differ country by country, so a frequency that is fine at home can be outside an allocation elsewhere.
Getting permission to operate
How a visitor may operate depends on agreements between countries. Three broad cases:
Situation
What it usually means
Agreement exists
The countries recognise each other's licenses. For US amateurs, CEPT covers many European countries, in both directions, and IARP (Inter-American Permit) covers certain countries in the Americas.
No agreement, but a procedure
The host country will issue a temporary or reciprocal license, sometimes for a fee, sometimes after weeks of paperwork.
Nothing available
Some countries allow no foreign operation. Do not transmit.
CEPT needs no /CEPT call sign suffix and no local-language ID.
Under CEPT you need to carry a copy of FCC Public Notice DA 16-1048, and you do not add "/CEPT" to your call sign. You identify in the way the host country requires, which commonly means the host's prefix before your own call (for example DL/W1XXX). Check the host's rule before you go.
Practical points
Look it up first. Start with the host country's amateur radio society or regulator, and the Callbooks and lookups pages for its prefixes. Rules change, and agreements are updated.
Keep to the host's rules, not only your own: bands, power, modes and what you may say. With foreign amateurs, keep to topics incidental to amateur radio and personal remarks.
Bring your paperwork: license copy, any permit, and a note of the rules you checked.
Equipment is a separate issue. Customs rules for radios, and any local approval of transmitters, are a different regulator from the amateur service.
At sea: operating from a US-documented vessel in international waters needs only your US license and the master's permission. Rules inside another country's waters follow that country.
Visitors coming the other way follow the same logic: CEPT amateurs from many European countries may operate in the US under their own licenses.