The term gain stages basically refers to all the points in an audio set-up (either for recording or doing live sound) where something can be turned up. And if it can be turned up, that means it cam possibly distort, which we will state for the record is bad - in any case where you aren't doing it on purpose for some effect.
The trick is to make sure you have enough signal at every stage but not too much, or you get distortion. But also not too little at any stage, or the stage after it will have to be turned up too loud to compensate, which can increase noise like hiss and hum.
Let's look at an example. In a live sound situation (where a band's microphones and instruments are all going through a mixer and PA system) an electric guitar's first gain stage is the volume knob on the guitar itself.
Next in the chain is the amplifier, which also has volume and gain knobs. Next comes the mixer, which will have at least two gain stages - the input gain/trim and the master output volume. But there is also the channel strip fader, and if you run the electric guitar channel through any effects, there is another gain stage. And some PA amplifiers and speakers also have volume controls, so there is another couple of gain stages.
So in order to get the best sound out of any audio source in both recording and live sound, you want to make sure you have a healthy signal level, but nothing that will cause any gain stage to be so loud that any of the audio distorts or clips anywhere in the signal chain. That process of ensuring the proper amount of signal without going over the "red line" anywhere is called "gain staging."
In recording vocals, the main goal is to make sure your microphone sends the right signal level to the preamp (this can be adjusted on some microphones by a pad switch), which then sends the right amount of signal to the recorder or computer. Most audio interface units have a knob to control this level.
Here's a video from Wink Sound that shows you an example with an electric guitar in a computer recording set-up.