Short version:
dBm... RF Power on the log scale simplifies calculating power vs. range (ignoring obstacles). From physics, signal strength will halve with doubling the distance. Thankfully this is true, which is why we can communicate with, for example, the Voyager spacecraft. It's also why cell phones are not a hazard to human tissue, especially CDMA (Verizon/Sprint and now, LTE).
dBi ... is a *relative* measure of antenna gain. 0dBi = gain of an ideal spherical antenna pattern. An antenna with directionality (non-spherical) has some gain, stated in dBi. A common 1/4 wavelength vertical is about 2-3dBi gain.
dBm is the common measure of absolute power in watts, relative to a standard of 1 milliwatt (0.001 watt). 0dBm == 1mW. 10dBm == 10mW. 20dBm == 100mW (0.1W). In RF transmitters, you need to clarify the claimed power in real world power-bandwidth, i.e., how wide of an RF channel is needed to encompass x percent of the transmitted power. There are government (in the US, it's FCC) regulations on this, to prevent spilling power out of the unlicensed 802.11 band. And there are WiFi standards for in-band adjacent channels.
dB, alone, is dimensionless. But 3dB is twice-again, no matter if it pertains to RF or sound.
I always mess up the unit designation.
As for gain, ideal quarter wave length IIRC is around 1.8dBi, half wave length is 2.15dBi.
Signal strength or power is reduced with the square of the distance. Field strength reduces linearly with distance.
Also to talk with voyager we use massive radio telescopes. When Galileos main antenna failed to unfold and we had to communicate using the backup, it requires some of the most powerful radio telescopes to talk to it at a very, very low signal rate (around 1kbps with compression, 160bps native IIRC). It was an isotropic antenna, for the low gain, and has a radio power of 15-20 watts. RSSI on Earth was -170dBm, or 10 zeptowatts (1x10^-21). The deep space network was used to communicate with it...using things like 35m radio telescopes. Those telescopes have a signal gain of 80dBi and a pointing of roughly 1/10,000,000th of the sky (or around 1/1th of an arc second).
Wireless is hard. By comparison if the failure had occured in Lunar orbit the radio power would have been around -138dBm and likely would have been able to be picked up with just a lowly 1-2m parabolic dish or so.
Voyager is significantly further away than Galileo was, but Voyager also has its high gain antenna pointed at us, at also is communicating at, I think, an even lower data rate (a few dozen bps at this point). It had a max, at Jupiter, com rate of 112.5Kbps. It also has a 12ft HGA. I am not sure the exact signal strength gain on it, but I'd guess at least in the 40-50dBi range...which means it was MASSIVELY better able to punch a signal to Earth compared to Galileo (and despite Voyagers current >>>10x distance greater from Voyager to Earth than Galileo to Earth).
Apollo used something like ~100w radios and resonably large sized parabolic (2m?) VHF/UHF antennas for radio and data links to Earth just from Lunar orbit and the surface of the Moon.
So, anyway, yeah I like using the wrong units.
dBi is antenna gain. dBd is also antenna gain, but normalized to a half wave length dipole. So 2.15dBi = 0dBd. The i is for Isotropic and the d is for Dipole...basically what the gain is normalized to (a perfect dipole or a perfect isotropic antenna. There can also be a negative gain on an antenna if the design is crap, its outside the radiation pattern, its high impedence or too low impedence, etc).
dBm...that is inverse square law. So doubling the distance would mean a reduction of 6dB (1/4th the received power). So -50dB at 30ft would be -56dB at 60ft and -62dB at 120ft. At ~1,000ft it would be -80dB. Hence long distances needing very high gain antennas.
Of course at 1,000ft if you added in a 8dBi antenna on each end over just a 0dBi (isotropic antenna) on each end, that -80dB becomes -64dB. Make it a pair of 12dBi antennas and it is -56dB...a pretty fair signal strength.