The P stands for phonogram, the legal term used in most English-speaking countries to refer to works known in U.S. copyright law as "sound recordings".
A sound recording has a separate copyright that is distinct from that of the underlying work (usually a musical work, expressible in musical notation and written lyrics), if any. The sound recording copyright notice is a copyright for just the sound itself, and will not apply to any other rendition or version, even if performed by the same artist(s).
In the U.S., the sound recording copyright notice, which may only be affixed to a phonorecord, consists of three elements:
the ℗ symbol;
the year of first publication of the sound recording; and
an identification of the owner of the copyright, either by name, abbreviation or other designation by which it is generally known. The identification can be omitted if the owner is the sound recording's producer, and the producer is identified on associated packaging.
The symbol in Unicode is U+2117℗sound recording copyright (HTML: ℗), with the alternative names "published" and "phonorecord sign". It closely resembles U+24C5Ⓟcircled latin capital letter p (HTML: Ⓟ) and U+24DFⓟcircled latin small letter p (HTML: ⓟ)
The symbol was introduced in 1971, essentially simultaneously in both domestic US law and international treaty.
In the United States, it was added by the Sound Recording Amendment (SRA), Pub. L. No. 92-140, § 3, 85 Stat. 391, 392 (1971). This statute was enacted October 15, 1971, effective February 15, 1972. It amended the 1909 Copyright Act by adding protection for sound recordings, and it prescribed a copyright notice for sound recordings:
(c) In section 19, title 17, of the United States Code, add the following at the end of the section: "In the case of reproductions of works specified in subsection (n) of section 5 of this title ["Sound recordings"], the notice shall consist of the symbol ℗ (the letter P in a circle), the year of first publication of the sound recording, and the name of the owner of copyright in the sound recording, or an abbreviation by which the name can be recognized, or a generally known alternative designation of the owner: Provided, That if the producer of the sound recording is named on the labels or containers of the reproduction, and if no other name appears in conjunction with the notice, his name shall be considered a part of the notice."