A. There are three sources for Hotza'ah:

1. The basic rule that all 39 Melachos that were in the building of the Mishkan are forbidden on Shabbos, as taught by a Halachah l'Moshe mi'Sinai. (The specific source for Hotza'ah is that the Leviyim moved boards from the wagons to the ground as part of the construction of the Mishkan.)

2. The Gemara in Shabbos 96b. The Gemara there cites the verse (Shemos 36:6): "Moshe commanded... Do not do more Melachah... and the people stopped bringing," which is interpreted to mean that they brought from a private house to the public area of the Leviyim, and from here the Melachah of Hotza'ah is derived.

3. The Gemara in Eruvin 17b cites the verse (Shemos 16:29): "A man shall not leave his place on the seventh day," which is interpreted to mean that he may not carry on him objects.

There is much discussion about how to understand the three sources, and whether they are independent of each other or complementary.

B. Toldos: Throwing an object from place to place, and moving by hand without the body moving (Rambam, Hilchos Shabbos 12:10).

C. Practical applications: Carrying in a pocket, including keys; carrying gum in mouth when walking outside; carrying from a private house to a jointly-owned yard or stairwell (the circumstances will determine whether forbidden mid'Oraisa or mid'Rabanan).

All the best,

Reuven Weiner