What is required for someone who holds a pilot certificate in one country to fly in another country? First, I'm wondering if there are any general rules that would apply to any such situation. Second, for the sake of a specific example, what would be required for someone who holds a private pilot certificate in the U.S. to fly an aircraft registered in the Philippines within the Philippines (assuming that the aircraft type is one for which the pilot is rated in the U.S.?)
For your specific example, Philippines Civil Aviation Regulations Part 2 (Personnel Licensing), section 2.2.4 covers "Validation and Conversion of Foreign Licenses and Ratings." It states that a holder of a license from a "Contracting State in accordance with ICAO Annex 1" (this includes the United States) is eligible for "validation" of their foreign license. The validation process involves presenting the appropriate documentation of your license, ratings and logbook. For more than just PPL privileges, you must also pass a "skill test" and demonstrate the required knowledge and ability. The conversion ("validation certificate") gives you the same ratings you had on your US license and is valid for 1 year.
The FAA has a very similar policy for foreign-to-US license conversion, covered in 14 CFR §61.75. As in the Philippines, the FAA does not require a skills test for private pilot privileges, but a written knowledge test is required for conversion of an instrument rating.
Another question here on ASE discusses conversion from EASA (Europe) to FAA (US).
Most other ICAO states recognize each other's licenses in a similar manner. In all cases you have to present your license and proof of experience, and in some cases pass a written test and/or checkride. I haven't look up the exact regulations for, say, EASA (Europe) or CAAC (China) but I'm sure they aren't far off from the examples above. In reality diplomatic considerations probably also come into play: a holder of an Iranian license would probably run into problems attempting to convert/validate their license in Israel, even though both are ICAO-contracting states, since they aren't on very good political terms to say the least. But this will likely be the exception rather than the rule.
(There are only 3 non-ICAO states: Dominica, Liechtenstein and Tuvalu.)
It varies hugely depending on the country. Iceland simply allows the holder of an ICAO license to exercise the privileges of a PPL for up to 90 days with no additional tests or validation required aside from a current medical.
I did this myself a year ago, and was able to determine the requirements by simply emailing the ICAA, who responded promptly:
Foreign visitors in Iceland can use their valid and current ICAO pilot licence for three months for private VFR flights in Icelandic airspace in aeroplanes registered in Iceland. So as long you as licence is current and valid and you have sufficient privileges to fly the aeroplane in question. You can fly private VFR for three months in Icelandic airspace without having to validate your licence.
This is a difficult question to answer without country specifics, unfortunately.