When I was involved with an evangelical church/group, the term was used to mean that God basically let you know that a particular path was either the way to go or not the way to go. For instance, someone could be praying about whether they should adopt a child. And, maybe that weekend at service, the pastor spoke on how we are all adopted children of God's. And then, maybe while listening to the local Contemporary Christian station, there was a commercial for a local Christian adoption agency. And then, to top it off, their dh came home that evening and told the story about how a co-worker was beginning adoption proceedings. That person would feel that they were being convicted by the Lord in terms of adoption being the correct path to pursue.
(I don't pretend to speak of all users of the word in a Christian sense. But, in my experience over the course of many years, that was how it was used, and how I still hear it used when I am around evangelicals. I also don't pretend to know if it were strictly an evangelical term, but since that is the only Christian circle in which I have associated, I do not know of whether the term is used outside evangelical groups.)