AndrejaKo's answer looks good but you specifically requested a layman's answer so I'll try to make it even less technical.
Firstly you are correct, there is no simple way to go from dB to range. Anything to do with radio involves far to many variables to give a simple 1 number answer for anything even close to real world usage.
However all things being equal you can apply some approximate rules of thumb.
For conventional radio signals a 3dB increase will give you a ~40% increase in range, 6dB a doubling of range.
For RFID type applications where signals have to bounce back to the reader again this effect is halved, 6dB will give you 40% more range, 12dB a doubling.
dBm is simply a measure of the power output. It is possible to convert this directly into watts if needed. 9dBm = 8mW, 30dBm = 1 Watt.
dBi is used to measure the antenna gain which is also indirectly a measure of directionallity. This is always a trade off.
If we assume the antenna is perfect then it will transmit all of the energy it receives from the electronics (a reasonable rough approximation for a good antenna). If it sent that energy equally in all directions the antenna would have a gain of 0 dBi, this equates to a gain of 1.
If you want to get more signal energy to go in a specific direction then you need to reduce the energy going in the other directions. If an antenna sent energy in a perfect hemispherical pattern it would have a gain of 3dBi.
Real world antenna gain patterns are far messier than the ideal but if an antenna has a gain of 7dBi then it's probably got a region about 90 degrees wide in which it transmits most of it's energy. That doesn't mean it doesn't transmit anything in other directions but the signal will be a lot weaker outside of that main region.
Why use dB for this sort of thing when we could use Watts for power and a normal linear gain number for the antenna? Three reasons:
- Firstly to confuse people, it's job security for engineers.
- Secondly the power levels involved in RF can vary massively, using dB
makes it a lot easier to deal with combinations of uW and kW in the
- Thirdly engineers are lazy and it makes the maths
easier. When working through total system power you can simply add
the dB numbers up (which are normally only 2-3 digits long) rather
than multiplying numbers which are different by orders of magnitude.
Some of those reasons may not be entirely true.
--- addition ---
For 900MHz RFID there are two different factors that can limit the range. Whether the tag is receiving enough power to switch on and if the tag is getting enough power whether the returning signal is strong enough for the receiver to pick up.
If getting sufficient power to the tag is the limit then the maximum range follows the inverse square law and a 3dB increase in signal will give a 40% increase in range, a 6dB increase will double the range.
If reading the reply is the limiting factor then the range limit is 1/power^4. A 3dB increase will give a ~19% increase, 6dB a ~40% increase and 12dB will double the range.
Which one of these two is limiting the range depends upon the tags and reader used. Older tags needed a fair amount of power and so the first of those two limits, the tag powering up, was often the limiting factor. Newer tag silicon is far lower power and so it's less certain what the limit will be these days.
When I indicate an increase in dB that can come from either source, if you change both then add the two changes together to give the total effect. If the increase comes from antenna changes then keep in mind the dB value is only valid for the direction with the maximum signal strength, that direction could change if the antenna changes.