For an Audio amp such as the LM 380/384/386, what is the power rating of the speakers they can handle? What is the logic behind selecting an amplifier and a speaker to work together as a unit? The LM 380 is rated as 2.5 W but it obviously drives speakers rated at much higher powers as I myself used a 10W 8 ohm speaker with it.

 

It is best to use speakers that have higher power handling capacity the the amp can put out. That way you do not damage the speaker.
And the impedance of the speaker is important so that you do not overload the amp with a too low value.
The LM386 recommended minimum speaker Impedance is 4 Ohms so if you were to use a 2 Ohm speaker it would stress the amp and not run to full power. Some amps it could actually damage.

 

The power rating of a speaker is simply the maximum power it can handle.
There's no problem with using a low power amplifier to drive a high power speaker.
The efficiency of the speaker is important if you have a low power amp.
Obviously for that, you want a high efficiency speaker (unfortunately the speaker efficiency is not always stated).