When we describe power in dB (decibels), it is relative to some reference level, like 1 Watt (W) or 1 milliWatt (mW). If the power in dBW is positive, it means it is greater than 1 W. If the power in dBW is negative, it means it is less than 1 W. And if the power in dBW is 0, it means the power is 1 W. Similarly for dBmW: if the power in dBm is positive, the power is more than 1 mW.

It's a bit like per cent in that respect. For example, 1.1 W is 10% up from 1 W, and 0.9 W is down 10% (i.e. it's -10%) from 1 W. However, decibels are not the same as per cent!

decibels are actually based on logarithms. The dB measure of power is 10 log (P/Pref), where P is the power and Pref is the reference power level. Suppose 1 W is the reference, and you have a 10 W signal. In dBW, that is 10 log(10/1) = 10 dBW. On the other hand, a signal of 0.1 W is 10 log(0.1/1) = -10 dBW.

A common rule of thumb is that doubling the power of a signal adds 3 dB, and that halving the power subtracts 3 dB. To see this, consider a signal of power P Watts, or 10 log(P) in dBW. Now double the power to 2P. In dBW, it is now 10 log(2P) = 10 log(2) + 10 log(P) = 3 + 10 log(P). This is approximate, but close.

Let's look at the specific values you mentioned. A power of 13 dB is 20 times the power of the reference level, so it's 20 W if you are using dBW, or 20 mW if you are using dBmW. Similarly, a power of -13 dB is 1/20 Watt (0.5 Watt) or 1/20 mW, depending on whether it's dBW or dBm.

Finally, an example. Some wireless personal area networks under development have restrictions on transmitter power. In any 3 kHz bandwidth, they are limited to +8 dBm. That's more than 1 mW and less than 10 mW. Taking the antilog, we see that the limitation is about 6.3 mW in any 3 kHz.