Archaeologists Do Not Dig Round Holes.
Archaeologists always make sure they dig rectangular holes. What they are trying to uncover is usually buried underground and must be uncovered very carefully. A rectangular or square hole allows them to remain organized and maintain a grid system.
This grid system is necessary to ensure that anything that is uncovered can be traced back to its original location after it is taken out. By breaking sections of the ground into measurable squares, it will be easier to determine the location and position of an object on the site relative to other objects.
When an object is uncovered, archaeologists must record it and measure it. They must also note other important factors such as its position, condition, and depth when uncovered. Archaeologists must go for accuracy to ensure that they will only be working with correct information for analysis and testing.
A garbage dump used by an older civilization is referred to as a midden. The type and quantity of garbage that older civilizations threw away holds plenty of information about the way they lived - their diet, lifestyle, culture, traditions, beliefs, etc. For an archaeologist, a person's garbage says a lot about him/her.
What archaeologists actually look for are clues and these include other things aside from pottery and tools. They look for animal bones, seeds, paintings, sculptures, structures, foundations, and even different types of soils.
As popular as he is, Indiana Jones does not do what a good archaeologist does and that is to preserve evidence. He barely takes notes (if at all) or records his discoveries, and often destroys artifacts. He does have amazing adventures, though.
The "finders keepers" rule does not apply... not always.
Even if they discovered gold, silver, precious gems, money, or anything of value, archaeologists are required to report it and hand it over to the authorities. These "authorities" will depend on which country or region the artifacts are dug. Essentially, archaeologists must adhere to the prevailing laws in the area. In Sweden, for example, anyone who discovers a single object of cultural value does not have to report it unless it is made of copper alloy or precious metals, at least partly. If there is more than one object, the find must be reported and the government will reward the finder.
In the U.S., all excavations are covered by the Archaeological Resources Protection Act and the National Historic Preservation Act. All excavations also require a permit. As part of the field's ethics, professionals do not keep what they discover. They also do not trade or sell because they believe that these objects belong to the public.
All digs require the go signal from the state archaeologist.
In the U.S., the state archaeologist must give his/her permission to any archaeologist who wishes to excavate a site in the state. It does matter even if the archaeologist is world famous.
Black market more than just destroys artifacts.
Every artifact found does not always make it to a museum. Some go to a private collector, sometimes, illegally. Because there is demand for artifacts, supply is kept flowing by black market traders, unethical professionals, and robbers. In some countries, robbers dig up some sites to steal artifacts for small amounts of cash. They often do not realize that they have done more harm than good, since these artifacts may hold new information about a culture that could be lost forever if not recorded and analyzed. Sometimes, the demand is so great that some people resort to faking artifacts to sell.
Archaeologists don't just dig.
In many cases, an archaeologist already knows which area to excavate. This is usually based on the history and other types of information gathered about the area including maps, historical writings, and records. Knowing what and where to dig also help archaeologist estimate the depth of the site. Of course, they also find surprises along the way.