I have heard it said that 'A miracle is when God chooses to be obvious, and a coincidence is when God chooses to remain anonymous,' but I would argue a third category of occurrences: not as grand as a miracle, but not as subtle as a coincidence. You could perhaps call it a help, a hint, a poke, a prod.
It is definitely when you can feel or you know the hand of God to be working in your life. He speaks to you, and you can almost hear His voice.
I think this is a purpose of the Holy Spirit--to give us access to the voice of God--for I have not heard of nonbelievers ever experiencing this; if they do experience it, they aren't aware of it until after they become believers. I have heard stories of times when something happened to someone when they were an unbeliever that they could not explain at the time, but which makes so much sense to them now that they believe. This is a purpose of the Holy Spirit--to make sense of spiritual things.
Recently, there have been many of these occurrences--these nudges from God--in my life, as well as in the lives of those around me. I cannot call them coincidences. I cannot call them miracles--at least, not as I define them: being visible to all, believer and nonbeliever alike. These occurrences are visible to believers alone. They are the every-day proddings of the Spirit. He is keeping in constant conversation with me; showing me that He is here with me.
It is a conversation. Except that when God talks, it is never idle chatter.