"But Rabshakeh said unto them, Hath my master sent me to thy master, and to thee, to speak these words?
Hath he not sent me to the men which sit on the wall, that they may eat their own dung, and drink their own piss with you?"
-- II Kings 18:27
Piss has long been a synonym for urine, although being slang it is somewhat more versatile than its more formal counterpart. Piss can be used as a noun, a verb, or an adjective. Historically it was not a 'bad' word; it has been used by doctors and poets, not to mention Biblical translators. However, in modern times it has become a borderline obscenity, and is not generally used in polite company or formal writing.
"Þe ʒerde... haÞ two holis, oon Þat Þe pisse passiÞ by and it is Þe ouer, and anoÞir Þat Þe sperme passiÞ by & it is Þe lowere."
The penis... has two holes; one that the piss passes through, and that one is above, and another that the sperm passes through, and it is the lower.
-- Surgical Treatises in Wellcome, 1392.
Piss comes to us from the Latin pissiare, which is thought to be onomatopoeic. Due to its ancient origins, some form of 'piss' occurs in most Romance languages, and in those languages it is nearly always considered obscene. The word pisser probably comes from a back-formation from piss, rather than directly from the Latin, but it's interesting how little this piece of slang has changed in over 500 years. As a side note, 'urine' also comes from Latin (urina), and also appears in many European languages; as in English, urine is the polite alternative to piss.
The less offensive word pee comes to us as a euphemism for the word piss (from the letter P), first appearing in 1902. The baby-talk form, pee-pee made it into print by 1923. Pee and pee-pee are currently used primarily by children, and have no obscene overtones. Whether or not piss is considered an appropriate word for the ears of children depends on where you live, but it is generally considered a no-no in filial settings.
"Monster, I do smell all horse-piss; at
which my nose is in great indignation."
-- Trinculo, The Tempest IV.i, William Shakespeare.
The word piss is used in any number of English idioms.