I would say that a supernatural encounter is anytime the supernatural overlaps in with the natural. The type I'll talk about is simply God encountering us.
To list a few that I've seen in my life... instant healing (swollen knees returning to size, an arm growing back, deaf ears hearing, cancer disappearing, and more), demons casted out, angel sightings, etc.
I'm sure that when you first read that paragraph there are many reactions going on
1) I'm a liar
2) i'm about to go on a charismatic tangent
3) maybe you're saying praise God
Though I've seen many of what we call the "big" miracles, I want to talk about something different. We too often overlook the miracles that occur around us everyday. Every person reading this has had a miracle today... you breathe, you woke up. The truth is you don't even rise from your sleep unless the Lord allows you too, so congratulations today the Lord wanted you alive and awake! LOL.
How about something as simple as peace from God. When there is chaos and hopelessness around you and all you have is peace. God gave you that peace, He came in an supernaturally encountered your world.
Of course the field of answered prayers is a whole other arena when God comes in and supernaturally or even sometimes naturally by using natural things, changes a situation or person.
I could really talk about this forever, but we need to remember that our God is a God of encounter. He created you for relationship with Him, so it would be foolish to think that He'd never want to touch your life, that He'd never want to intervene, or communicate with you.
I think we get too caught up in our day to day life and we lose the fascination of the Creator of the universe being available to you. Available to communicate with, available to cry with, available to make requests to, available to lean on, available to give you that miracle you need, available to love you and give you the strength and peace you need.
We serve a God who desires to encounter us are you listening, are you watching, do you even want it?