Perhaps the most common to the public is people per square mile (ppsm) or kilometer. This is a great one for measuring a region or a certain area, especially for comparison purposes. New York, for example, has a population of 8,175,133 over 305 square miles (8,175,133 / 305 = 26,804 ppsm). Los Angeles population is 3,792,621 contained in 469 square miles for a ppsm of 8,087. Dallas' ppsm is 3,111 which is the population 1,197,816 divided by a land area of 385 square miles.
This works on a macro scale. Generally speaking, the higher the ppsm, the greater the urbanity. A look at the list of top five densest major cities reveals: 1) New York, 2) San Francisco, 3) Boston, 4) Chicago and 5) Philadelphia. If I were to make a list of most urban places, those five would certainly be in the upper tier.
The main caution in using this measurement is the boundary of the area in question. Some sprawl proponents will point out that L.A. is denser than New York. While that is true from a regional perspective (New York State has done a good job of preserving large portions of its countryside), the city is far more dense, and therefore far more urban than L.A. Transit use alone shows that to be the case (New York carries 12.8 million trips a day, L.A. has less than 1.5). As with any stat, a grain of salt and some perspective must be exercised.
A common measurement amongst developers and their development is the unit per acre. This is generally more suited for suburban, low density developments, like a single-family subdivision. Add the number of houses, divide by the number of acres and you get your number. A small development of ten acres with ten houses yields one. 40 quarter acre lots in the same development yields a four. It is also important to note, that unlike ppsm, units/acre does not consider infrastructure. That is a separate column for planners to deal with in their plans.
It can also work for urban developments, but often developers don't like those numbers to come out because NIMBY's will see it and automatically oppose it. Most downtown buildings would see that number well over the 100's.
This is also the most common measurement that NIMBY's use in opposing projects. They see the higher the number, the higher the externalities. In some instances that may be true. An apartment complex with a relatively high units/acre usually will generate a higher crime rate, traffic trips, pollution, but that is usually in the context of an auto-oriented area. Those cities in the top five densest generally have lower negative externality rates than their suburban designed counterparts per capita and in sime case straight up. Design and use can make a huge difference.
This measurement is in the middle on an area scale. It doesn't effectively apply to regions, but can and mostly is used above the individual property, like master-planned communities.
From an urban design standpoint, one of my favorite if the Floor-to-Area Ratio (FAR). It is also the most property specific. Reference the illustration as I explain this if you are unfamiliar.
From the property lines, if a developer builds a one-floor structure across the entire property, it's FAR would be 1.0, as the picture on the left shows. In the middle, the developer built the structure on only 50% on the property, but built two stories. This is also a FAR of 1.0. Now if the developer built three stories, it would be a FAR of 1.5 (.5 x 3 = 1.5).
For an urban area, it is optimal to have a FAR as close as equal to the number of floors as possible. This generally ensures better urban design and limits setbacks, which are a detriment to the street level and pedestrian experience.
Now there are several drawbacks to this measure, as with any other tool. It doesn't ensure that the streetscape will be pedestrian-friendly. It doesn't ensure it will have multiple uses. It doesn't ensure the users will engage in urban activity. The Empire State Building is a great urban building, but its FAR is far smaller than its floors because of the setbacks. Conversely, Harwood Center in Dallas has a near equal ration, yet is a sub-par urban structure.
When I look at downtown Dallas as a whole, I see a lot of buildings with a FAR that is half or less of the number of floors. Bank of America Plaza, Comerica Bank Tower, Chase Tower, Trammell Crow, Thanksgiving Tower and Energy Plaza are all in the top ten in height, but have drastically lower FAR's.
On the flip side, that isn't to say that a setback, and therefore lower FAR compared to floor height, are inherently negative. Lincoln Plaza has a pretty big setback, but at the street level, the space is used for greater sidewalks and pedestrian amenities. Conversely, they wouldn't need to do that if the sidewalks were wide enough and the city hadn't converted some of the space to give to cars for more traffic lanes.
Bottom line, Dallas has a downtown that looks dense from a distance, but when you get there, it seems empty. The big reason is the FAR is nowhere near as close at the heights appear. In essence, it is a false density. When critics say cities like Dallas can't support tall skyscrapers, this is what they mean. In an auto-based city, density doesn't work because cars need too much space for storage when you park. The more space in a building, the more parking you need. Many of those are surface parking lots, which have a FAR of zero, which is also their contribution to a vibrant street scene.