The whole story
The s/n ratio is the Signal to Noise ratio. Assuming your radio is making the normal hiss that it would with absolutely no signal, how much stronger is that signal from that usual background Hiss?
A 10 dB signal to noise ratio means that the signal is ten times louder than the background noise.
A 20 dB signal to noise ration means that the signal is one hundred times louder than the background noise.
You tend to hear things on an logarithmic scale, so it is customary to use a log of a ratio to describe performance in easy to understand terms.
The radio with the 1 microVolt at 20 dB signal to noise ratio is therefore considerably more sensitive than the radio with the 3 microVolts for a 10 dB signal to noise ratio.
HOWEVER: That's not the whole story. There is another specification which can appear in several different ways called dynamic range. Having super sensitive hearing isn't going to help you much when you're standing in the middle of a rock concert. Likewise, if you live in a city with lots of signal sources, it could well be that the more "deaf" radio actually may perform better.
It's like someone wearing earplugs at a gun range. You definitely won't hear the super quiet sounds you would hear without earplugs. However, if you don't wear them, your ears may start ringing from gunfire noise and you won't be able to hear what people say next to you anyway.
The subject of how one measures dynamic range is unfortunately quite complex. People literally write books on the subject. However, one easy to understand number is called Blocking Dynamic Range. The higher that number is, the better.
That's the short answer. Hope this helps...