On a GSM phone (AT&T, T-Mobile), the bars bear some relationship to 2G signal strength (but they're not a direct signal strength indicator). On a CDMA phone (Verizon, Sprint), they indicate something called signal quality, which is really a measure of how much the channel your phone is on is being used. You can get a perfect call with 1 bar and have all sorts of problems with 5 bars. The indicator is there more because people expect one than to actually indicate anything that would be of use to anyone except an engineer troubleshooting the network. (Even signal strength apps won't help much - a strong signal on an overloaded tower face will give you dropouts and "underwater, while a weak signal on a tower face on which you're the only user will give you a good quality call.)
The indicators were useful back in the days of analog cellphones, but as of February 18, 2008, carriers were no longer required to provide analog service, and most of them had already converted to digital and dropped analog. But people were used to "signal strength" indicators, so they're still on the phones, even though they're useless.
And that's for voice. The indicator has nothing at all to do with QoS (quality of service) on your data connection.