E.164 is the ITU-T standard that gives every phone number one unambiguous international form: a plus sign, a country calling code, and at most fifteen digits in total, with no spaces, brackets or dashes. +14155550142 is E.164; (415) 555-0142 is the same number written for a person.
What Spaw answers
e164 is the canonical form and the field to store. Three others carry the same number written differently: national for display inside its own country, international for display across borders, and rfc3966 as a tel: URI for a dial link.
Why the country parameter exists
Every region's calling code, with an example number in each form, is on its page under phone facts by country.
E.164 is unambiguous only when the calling code is present. A number typed without one — 4155550142, or a national form beginning with a trunk 0 — could belong to several countries, so a lookup needs country to know which numbering plan to read it against. Without one the answer is missing_country, which is free.
An API key can carry a default country, applied only when the request body names none at all; an explicit country: null still means "read it from the calling code".