I don't use 3DS max, so I can't give you specific direction, but I can tell you generally what's happening. I assume you were following some kind of tutorial, so you may not remember doing this. However, it looks like you have the model set up to be mirrored down the center. So, selecting one vertex will mirror to the other side. Typically this is very useful for creating symmetrical models like a face. However, if you want the model to be asymmetrical, it's not so useful. Here's a video showing the use of mirror.
I suggest keeping this mirror mode on until you're finished making most of the changes to your model. Then, when you're done with that, disable it and make your asymmetrical changes.
Sorry I don't know exactly how to disable it.