Bridge mode is special way of dealing with the inputs and outputs.
Many (read that as most) amplifiers are what you call single ended, ie they run between a certain voltage and ground (0 volts). Many of those take the supply voltage, split that in half, and then the amplifier can only go a certain amount above and below that 1/2 way point, pushing through an output capacitor, into the speaker, and out to ground. So your output is limited to a percentage of 1/2 the supply voltage. Yes, there are other amps that run + and – power supplies in order to get that much more voltage swing, but that’s another post.
Now, if you reconfigure things a bit by adding a second part to the input where you get both a “positive” (upward going to start) and “negative” (downward going to start) signal, then apply that to 2 of your power amplifier sections, connecting the output of the amps to the speaker + & – (no ground reference is used), you get twice the voltage swing as before. Say you had 4 volts RMS output before (single ended), now you have 8 volts RMS (bridged). What you have done is actually quadrupled your power output.
(RMS V)^2 / Impedance
(4^2) / 8 = 16 / 8 = 2 watts
(8^2) / 8 = 64 / 8 = 8 watts
(^2) is “squared” or “raised to the power of 2”
As for the impedance of the bridged mode being rated @ 2 or 4 ohm, you will be fine with 8 ohm, generally, since it is solid state (transistor) which has no real problem seeing greater and greater impedances. Granted, the higher the impedance mismatch, the more it affects the bandwidth (shrinks). But 8 ohm should be no problem. Just know that the power will be about 1/2 the 4 ohm rating.