What exactly does Watts and dB signify?
What are Decibel?

Decibel are a measurement of power. It can be calculated by the following formula:

$$a_{db} = 10*log(\frac{P}{P_0})$$

Decibel is a measurement that is relative, which means it always has a reference. Usually this reference is 1 milliwatt and the pseudounit is "dbm". The unit is a pseudounit because there can't be any units inside of mathematical functions such as the logarithm. 

There are several suffixes and all vary in their reference:

dbm is referenced to 1mW, so P_0 is 0,001W.  

dbW is referenced to 1W, so P_0 is 1W. 

db is referenced to some power level you measure somewhere. This is used for amplifiers to quickly determine what gain they have (Output power referenced to the input power).

Keep in mind that there are only measurements of power!

Now we often see units like dbµ and dbV. The assumption there is that P=U^2/R and if R1 equals R2 they cancel out (as do the voltages!):

$$a_  {dbµ} = 10*log(\frac{P}{P_0}) = 10*log(\frac{\frac{U_1^2}{R_1}}{\frac{U_2^2}{R_2}}) =10*log(\big(\frac{U_1*R_2}{U_2*R_1}\big)^2)= 20*log(\frac{U_1*R_2}{U_2*R_1}) \\ = 20*log(\frac{U_1}{U_2})$$

The reference would be µV and V.

Why do we use Decibel?

In communications engineering we often need to draw diagrams of signal strength and Bode plots for filters. Now imagine a usual GSM base station which is transmitting with 1Watt and therefore 30dbm. The attenuation is usually the factor 1000000000 (90db) or more. So your mobile has to work with signal strengths as low as 30dbm-90dbm = -60dbm which equals 0.000001mW.

Ten decibel is (thanks to the logarithm) an amplification/attenuation of one decade (multiplication/division by 10). That means you make huge numbers very small and you can draw a diagram of signal strengths or Bode plots for filters without going from 0.0000000mW to 100000000W. Another benefit is that we can add and subtract the decibel instead of multiplicating the numbers which is a feature derived from the logarithmic calculus:

100dbm = 10 Megawatt
-50dbm = 0.0000001 Watt

Your specific usecase

The wattage on your amplifier tells you how loud in reference to how silent it can be. The reference here is the noise-floor your amplifier has (every component is noisy! Even resistors!) in comparison to the maximum it can do. this is called dynamic range and calculated by 
$$10*log(\frac{P_{100\%}}{P_{0\%}})$$

The higher the dynamic range the higher is its output power or the lower the noisefloor.
Another thing that this tells you is the approximate power level your speaker should be rated for.

You are supposed to use a higher power amp when your audience is very big, eg at concerts. 4 times the power equals 2 times the volume (same speakers provided). 150Watt is a very big amplifier in a room if you have decent loudspeakers (not the cheap ones from the grocery store). A higher wattage also means that the amp will drain your battery a lot faster (this is linear. 100W amps the battery twice as fast as the same amp with 50W output). The speaker receives - of course - a larger amount of power and therefore displaces more. You can crank the volume up until the speaker or the amplifier is at it's limit (audible crackling noises).

All in all the listener determines  6db as approximately twice the loudness at which you need to provide 4 times the power to achieve it.

My opinion is that you rather should look out for a speaker with a high efficiency (the enclosure is also an important factor!) and a smaller amplifier.