The final melacha involves transporting things outside on Shabbos. When the Mishkan was moved from place to place, it had to be disassembled and the boards that formed the walls were loaded onto large wagons. The wagons traveled to where the Mishkan was to be re-assembled, where the boards were unloaded. The wilderness where the Jews traveled was a public domain (called a r’shus harabim in Hebrew); the wagons were of such a size that they constituted a separate private domain (a r’shus hayachid in Hebrew). Moving something between a private and a public domain or transporting something through a public domain is the melacha of hotza’ah, carrying. (Actually, carrying within a public domain is technically called ha’avarah, but we’ll refer to the entire package as hotza’ah.)
While we may refer to hotza’ah as “carrying,” it more literally mean “bringing out” and refers to any manner of getting an item from point A to point B. Throwing something from a private to a public domain, or rolling it through a public domain, is the same melacha as carrying it in one’s hand or pocket. (“Wearing” something that is not a garment is still carrying it, so no “book hats” allowed!)
A public domain has certain parameters in terms of size and traffic flow. Many areas we treat as public domains are not truly such according to the Biblical definition, but we treat them so by rabbinic enactment. These areas may be “fixed” for carrying by setting up an “eiruv,” which is effectively a series of doors. (The details for where an eiruv may be established and what doing so entails are beyond our scope.)
Hotza’ah is so important that it’s the reason we don’t blow the shofar when Rosh Hashana falls on Shabbos – so that we won’t come to carry the shofar to shul through a public domain.
The melachos of Shabbos are dealt with in the Talmudic tractate called Shabbos; hotza’ah is the very first topic. Hotza’ah is also the only melacha that has its own tractate, called Eiruvin. It should go without saying that this is an area that requires great study.
This is just an introduction to the concepts of the melacha of hotza’ah; it is not a substitute for a full study of the halachos.