Mechiras Chometz: How it Works
Chometz on Passover
The Torah (Ex. 13; 7), describes the chometz-ownership prohibition using two phrases:
"Chometz should not be seen for you"
and
"Sourdough (se'or) should not be found for you".
The Oral Torah teaches that the phrase "for you" mentioned in both these passages is to be understood as "belonging to you". That is to say, it is forbidden for a Jew to have chometz in his possession over Passover regardless of whether it is readily visible ("should not be seen") or hidden out of view ("should not be found").
The Torah (ibid 12; 15) further commands us to "destroy it", as though to teach that, in order to ensure that one does not come to a state in which chometz could be "seen" or "found" on Passover, a Jew is obligated to search his property for chometz and then destroy it.
Now, since the fundamental chometz prohibition (that it should not be seen or found) depends on ownership (i.e., if it isn't yours, its existence isn't prohibited), selling chometz to a non-Jew before Passover should remove the obligation to "annul and destroy" it.
Fulfilling the mitzvah to destroy chometz
But doesn't the Torah obligate a Jew to physically destroy his chometz? By selling it then, isn't he evading a positive mitzvah? Many authorities write that not only does selling chometz to a non-Jew remove the Jew's ownership of the chometz (and thus, his obligation to destroy it) but, in fact, it also is a fulfillment of his actual positive mitzvah to destroy it.
It should be noted, however, that the Chayai Adam (ch. 124, Nishmas Adam Sha'alah 8) writes that physical destruction is the only way to properly fulfill this mitzvah. Therefore, bearing this opinion in mind, it is preferable to sell most, but not quite all, of the chometz in one's possession, and, with the little you have left over, fulfill the mitzvah of "bi'yur".
Selling chometz through a rabbi
Originally, everyone would privately sell his own chometz to his non-Jewish neighbor, however, because of the complicated halachic issues and the many potential difficulties surrounding such sales, the custom arose among Jews of later generations to sell only through a qualified rabbi, a Jewish court or through an individual appointed to the task (concerning the evolution of this practice, see the essay of the late Rabbi Zevin in "Moadim b'Halacha" Pesach; Mechiras Chometz).
It is important to understand that one who sells his chometz through his rabbi, isn't actually selling the chometz to the rabbi, rather, he is appointing the rabbi as his agent in the sale (through a special "harsha'ah" contract - the form used to delegate power of sale to your rabbi). Signing this contract authorizes the rabbi to then sell the chometz to the non-Jew.
Respecting the Non-Jew's ownership
A proper sale to a non-Jew requires that the chometz itself is formally sold along with the sale or rental of all the places in which or on which the chometz rests.
Any place in which chometz (that's been sold) will be stored through Passover must be distinctly marked off from your other possessions (so that you shouldn't inadvertently eat or make any use of it - Shulchan Aruch Orech Chaim 440; 2). If creating a solid physical partition (like a locked cupboard or a sealed room) is not possible, you should at least cover the chometz with a layer of wrapping paper or something similar (and seal it with tape). In any case, you should leave some written warning sign on the outside.
If you rent your whole house (or one whole room) to the non-Jew, you should lock it up before Passover and not enter throughout the entire festival (Shivas Zion 10 - Seder Mechiras Chometz by the Gaon Nodah B'Yehuda).
Selling Investment Instruments
According to Cheshev Ha'Ephod (vol 1, #62 and #63), the chometz portions of common shares in publicly held companies are not considered the property of the shareowners in regard to the Torah prohibition of owning chometz on Passover.
However, it was the opinion of Rabbi Yakov Yitzchak Weiss ztz"l (Av Beis Din, Jerusalem), that shares which allow their owners management rights (for instance, the right to vote at annual general meetings) make shareholders partners in the business (see Shulchan Aruch, Orech Chaim #450; 3) and, the relative portion of any chometz owned by the company would, therefore, be halachically considered the shareowner's property (Minchas Yitzchak vol 3 #1 and vol 3 #26).
However, shares (like those of mutual funds) which don't carry any voting rights and provide only a portion in the company profits (or losses) are not considered the property of their shareowners for the purposes of Passover law. These, therefore, need not be sold with a Jew's chometz (Minchas Yitzchak vol 3 #1).