Choosing the Transmitted Power
All beacons allow you to adjust the transmitted power from, for example, -30dBm to +4dBm. dBm stands for Decibel-milliwatt, a unit used to measure radio frequency (RF) power level. dB (without the 'm') measures the power of a signal as a function of its ratio to another standardized value and the m in 'dBm' indicates we are comparing relative to 1 mW of power.
It helps to understand transmitted power. The following diagram shows how a given transmission radiates out such that you don't get half the power at double the distance:
The relationship actually follows an inverse square law and the logarithmic dB scale is used to represent this such that each +3dB change represents a doubling of power. -3bBm is a halving of power. +10 dB is ten times the transmit power.
So, for the beacon, 0dBm is the standard power level (we'll cover 'standard' later) and +4dBm is just over a doubling of power. -3dBm will be half power, -6dBm will be a 1/4 power and -9dBm 1/8 power. The power level affects battery life. At higher power levels, transmission is the main use of battery power and hence every +3dBm will very very roughly flatten your battery in half the time. It's 'roughly' because the beacon circuit also uses power to run itself that isn't to do with transmission. At very low transmission power, the beacon circuit battery use will become more significant and halving the battery power will usually (it depends on the beacon) get you less and less proportional gains in battery life.
Transmitted power affects the range. However, doubling the power (+3dBm) doesn't double the range due to the previously mentioned inverse square law. Increasing power actually has less measured affect at the position near the limit of the range.
The most common reason to change the transmitted power is to influence battery life or to change the range to purposely make it cover a very small or a very large area. The article on choosing Choosing UUID, Major, Minor and Eddystone-UID For Beacons also explains how you might want to decrease the transmitted power to prevent iOS regions from overlapping.
A stronger signal also produces less error when determining distance from the received signal strength (RSSI). This is because, electrically, low signal levels are nearer those of noise. i.e. The signal to noise ratio is higher with higher power. Scenarios that depend on more accurate inferred distance use more transmitted power even though the extra range might not be needed.
We mentioned the 'standard' power level. Some beacons (configuration apps) allow you to change the Measured Power which indicates the expected received signal strength at a distance of 1 meter from the beacon. Changing this, doesn't change the power. It's only a calibration value. It manifests itself in the advertising data as a value that the receiver (OS or receiving app) can use in a calculation to determine distance based on RSSI. You shouldn't usually change this value. It's used for fine tuning some limited scenarios to take account of obstructions or other such peculiarities of the physical beacon placement.
Finally, remember also it's possible to choose USB powered beacons in key places to transmit higher transmitted power without the detriment of lowering battery life.