Memory saturation is when your VPS is using up its entire allotment of memory. When this happens, a variety of things will start to happen. Your sites will likely stop responding in many cases, but beyond that other essential processes on your VPS will stop responding as well such as the SSH server, FTP server, streaming media server, and so. It's entirely possible for your VPS to enter a state where you can't even log in due to memory saturation.
Checking for the cause
You can usually tell if this is happening to you by checking a few things:
Checking your resource graph in the panel
If the resource usage graph is showing your memory spiking high above the memory allocated, then this is likely what's happening.
Run the 'free' command'
If you are still able to log into your VPS via SSH, then try running the free -m command. This shows you how much free memory you have available. The closer that is to zero, the worse you are. At zero, you've reached total memory saturation, and what happens next can be unpredictable.