If one sees an item that belongs to another Jew and was lost, he is obligated to take that object and return it to its owner. One who sees a lost object and ignores it violates a Misvat Lo Ta’aseh (Torah prohibition) that forbids ignoring a lost object, and also neglects a Misvat Aseh (affirmative Biblical command) to return lost objects to their owner, as the Torah commands, "Do not see your fellow’s ox or sheep wandering astray and ignore them; you shall return them to your fellow" (Debarim 22:1). If a person takes the item with the intention of keeping it, rather than returning it to its owner, he also violates the Torah prohibition against theft ("Lo Tigzol" – Vayikra 19:13). If, however, the finder then decides to return the object to its owner, he has corrected his mistake and is no longer in violation of these prohibitions, assuming the owner had not despaired from retrieving the item. If the owner despaired after the finder took the object for himself, the finder is guilty of a these violations even if he then returns the object.
This obligation does not apply to possessions that have no Siman – meaning, a distinguishing feature by which it can be identified as a person’s possession. Money, for example, has no Siman; there is nothing on a coin or dollar bill whereby it can be definitively identified as belonging to a particular person. Therefore, if a person dropped money in a public place, one who finds the money is allowed, strictly speaking, to keep the money. People generally feel their pockets regularly to ensure their money is still with them, and it can therefore be assumed that the money’s owner realizes he has lost the money and has despaired, since the money has no Siman and was left in a public place. According to the strict Halacha, then, one who finds money that was lost in a public area may keep the money for himself, even if he knows whom it belonged to, since the owner has presumably despaired.
Of course, this applies only if the money is indeed unidentifiable as a particular person’s property. If the money was found in a wallet, or it was stacked in some particular way whereby it can be identified as belonging to somebody, then the finder must return the money to its owner.
Although this is the strict Halacha, one who finds lost money fulfills a Midat Hasidut (measure of piety) if he returns it to its owner. He is allowed to keep the money, but it is worthwhile as a Midat Hasidut to return it.
The question arises as to why the Torah allows one to keep money that is found in a public area, even if he knows who lost it. Why shouldn’t the finder be required to return the money to its rightful owner? The answer, as mentioned in several works, is that Hashem "settles all accounts" and we can rest assured that the finder rightfully deserves the money. It is possible, for example, that in a previous Gilgul (incarnation) the owner owed the finder money but never paid him, and for this reason Hashem arranged that this money would be lost and then found. We never know for sure the precise reasons why these situations occur, but we do know that God precisely calculates people’s assets, and that there is a reason why He saw to it that a person would lose money and it would be found by somebody else.
(Based on Ben Ish Hai, Parashat Ki-Tabo)
Summary: It is forbidden to ignore an item that was lost by a fellow Jew and one finds, and it is certainly forbidden to keep it for oneself. However, if the item has no distinguishable feature, such as in the case of lost money, the finder may, strictly speaking, keep it for himself, though as an extra measure of piety he may wish to return it.