This two-player game has been played in America, primarily in Jewish communities, since at least the mid twentieth century, and presumably originated in Europe. There are many slightly different versions and spellings of the name - I have seen "Pisha Pasha", "Pisha Paysha" and even "Persian Pasha" as alternatives.
Pishe Pasha is usually played by two players, using a standard 52 card pack, but is is possible for three to play - see variations. The cards are shuffled and each player is dealt a stock of 26 cards, stacked face down. Traditionally the first cards are dealt two at a time, and the remainder are dealt three at a time.
The object is to get rid of all your cards, either by playing them to foundation piles, which are built up A-2-3-...-K in each suit in the center of the table (as in solitaire), or by playing them on your opponent's discard pile.
The non-dealer begins by turning the top card of her stock pile. If this card is not an ace, it must be discarded face-up to start the non-dealer's discard pile. It is then the dealer's turn.
If the non-dealer's first card is an ace, she must play it to start the foundation pile of that suit. The non-dealer then turns up her next stock card, which must also be played to a foundation pile if it 'fits' (being an ace or the next higher card of the same suit as the top card of a pile that has already been started). The non-dealer continues to turn up cards from her stock until a card is found that does not fit on any foundation pile. This card must be discarded face up, ending the non-dealer's first turn.
The dealer now plays, turning the top card of his stock pile face up.
The dealer continues turning up further cards from his stock as long as they can be played either on some foundation pile or the opponent's pile. When an unplayable card is turned up, the dealer must discard it to start his own face-up pile. The dealer's turn then ends, and it is the non-dealer's second turn to play.
From each player's second turn onwards, there is an additional option. At any point of your turn, the top card of your own discard pile can be played onto a foundation pile or your opponent's discard pile according to rules 1 and 2 above. If the top card of your discard pile can be played on a foundation pile, you must play it there before doing anything else, but moving cards from your discard pile to your opponent's discard pile is voluntary.
When you cannot or do not wish to play the top card of your discard pile, you turn over the top card of your stock and try to play that according to rules 1 and 2 above. This play may make further plays from your discard pile possible. Your turn continues, playing cards from your stock and your discard pile according to the above rules, until you turn up a card from your stock that cannot be played on any foundation, and which you cannot or do not wish to play on your opponent. You must discard this card on your own discard pile, ending your turn, and it is your opponent's turn to play.
Here is an example of a possible layout durung a game.
It is player A's turn. Since player A's top discard does not fit on B's discard pile or any foundation, A must turn a stock card. If this card is the 
4, 
7, 
2 or 
A it must be played on the appropriate foundation pile. Note that the 7 cannot be played on the opponent's 8 - it must go on the 6. If A turns the 7, it can be played on B's discard pile, and then the 8 can be played on top of it. This may allow a series of cards from A's face up pile to be given to B, if the top cards of A's pile are in sequence, as they will be if they have just been placed there by B.
If a player breaks the rules of play, his opponent may call "stop", and then has the option of either forcing the correct play to be made, or of causing the offending player's turn to end, and starting her own turn. The most common error is failure to notice that the top card of your discard pile has become playable on a foundation - if you turn the next card of the stock in this situation your opponent can call "stop". Another cause for a stop call would be playing a stock card on your opponent instead of on a foundation pile.
If a player misses an opportunity to play a card to the opponent's discard pile, there is no penalty. This is legal and in some circumstances it is better not to give your opponent a card. However, once you play a card to your own pile and let go of it, you forfeit the chance to play it on your opponent's pile if it fitted. Also, once you turn a card from your stock, you must deal with that card before you can move any card from your own discard pile to your opponent's.
When your stock is empty and you are ready to draw another card from it, you turn over all the cards of your discard pile (as a block, without disturbing their order) to make a new stock. Your discard pile will then be empty until you discard a card there at the end of your turn.
If you succeed in emptying your stock pile and your discard pile, you have won! The winner scores one point for each card remaining in the opponent's stock and discard piles.
Although this is a simple game, there is some strategy to it. Always try to place the highest cards possible on your opponent's discard pile during the game. To be left with kings in your stock is not good. Also, it is beneficial to remember the order of the cards in your face-down pile - this helps you to decide when to stop laying off cards onto your opponent from your discard pile and turn a stock card instead.
Some play without the stop rule. A player who fails to play a card to a foundation can be forced to correct the mistake, but the player's turn continues.
Each player has a face-down stock of 17 cards and one face-up card is placed in the center of the table. This card starts a single foundation pile which is built upwards regardless of suit, and continues from king to ace, two, and so on.
Apart from this the mechanism is almost identical to the two-player game described above. The turn to play passes clockwise. At your turn you can play from the top of your discard pile or turn up the top card of your stock. It is compulsory to play to the centre pile whenever your top discard or the stock card you have just turned fits there. Otherwise you can play the next higher or next lower card on either opponent's discard pile, ace being considered as next to king and two. Failing that you discard on your own discard pile, and the turn passes to the next player.
It is also possible to play the two-player game this way, with a single foundation pile that is built upwards ignoring suit, continuing from king to ace to two. With two players each starts with a stock of 26 cards, there is no center card, and the single foundation pile starts from ace.
Several card game books describe a much simpler game under the name Pisha Pasha. This is a game of pure chance, similar to War. It is not clear whether this is a misunderstanding that has been copied uncritically from book to book, or whether this much less interesting game has a genuine following.
This simple game is also for two players, using a standard 52 card pack which they divide equally between them. Each player holds his half of the pack face down and, the players simultaneously turn over their top cards onto their own discard piles. If the two turned cards are of different suits nothing happens, but if they are the same suit, the person who turned the higher card (using the ranking from high to low A-K-Q-J-10-9-8-7-6-5-4-3-2) takes the whole of the opponent's discard pile and adds it to the bottom of her face down pile.
When your stock cards run out, you pick up your discard pile, turn it over and use it as a new stock. The game continues until one player wins by collecting all the cards. (Rarely the game can reach a stalemate in which the suits of the turned up cards never match. If this happens, the player who has more cards wins).