E.164
The international standard format for phone numbers: a plus sign, the country code, then the subscriber number, up to 15 digits.
E.164 is the standard that makes +14155552671 unambiguous anywhere in the world. A leading plus, the country calling code, then the national number, with no spaces or punctuation and never more than fifteen digits in total.
Interfaces, databases and messaging systems normalise numbers to this form before doing anything with them.
Reading one apart
The plus tells any system that an international prefix follows, so the number works from any country without a local dialling prefix. The country code is one to three digits, and everything after it is the national number as the destination country defines it.
National trunk prefixes are dropped in this form. A United Kingdom mobile written 07700 900123 at home becomes +447700900123, and a German 0151 number becomes +49151 followed by the rest.
When a form rejects a number you know is right
Formatting is the first suspect. Entering both the international prefix and the national leading zero is the most common mistake, and it makes a valid number look too long.
Type the international form with no spaces, or pick the country in the form's own selector and enter the national digits without the leading zero. The number lookup tool shows the correct international form for any number you paste into it.
Where the fifteen-digit limit comes from
The standard reserves fifteen digits for the whole number after the plus sign, with the country code counted inside that limit. National numbering plans are built to fit, which is why no legitimate number runs longer.
Systems that store numbers as integers or trim leading characters break on this regularly. Keeping the plus sign and storing the value as text avoids the whole class of bug, and it is why interfaces ask for the international form.