How to install a new pvc line on a fridge to an existing small plastic line?
Most refrigerators come with either a 1/4" plastic or copper tube or a male hose thread (just like your garden hose spigot) inlet.

If you are connecting 1/4" tube to 1/4" tube, you need a compression coupling:
 

The inserts slide into the plastic tube to prevent it from collapsing when you tighten the fittings.

If you have a hose thread connector you will need an adapter:

The picture does not show the hose washer, you need that too.