As others have pointed out, it isn't just with believers that you can sometimes sense things like that. Sometimes the Holy Spirit is nudging you or bringing attention to something regarding someone else.
Just the other day, I crossed paths with someone and my soul reacted strongly to them even though we barely interacted. I tried to discern what was going on, and for various reasons came to the conclusion that it was the Holy Spirit warning me to keep my distance.
Also, with a close friend (who isn't saved yet) the first time we met it was like the Holy Spirit pointed to him and said to me "See? He is the reason I have you at this miserable workplace. God has a plan for his life and you're a part of that plan." I didn't hear that audibly, of course, but answered prayers and events that followed have convinced me that I heard correctly.
How do we describe the voice of the Holy Spirit? Like a gentle whisper? An intruding thought? A certain feeling? Early Chinese Christians called the Holy Spirit "the Cool Breeze"... It's something that's more or less impossible to explain to unbelievers. They want physical evidence for everything, but the Holy Spirit is spiritual and almost never in physical form. So to the unsaved it sounds like we're just describing our own emotions or our conscience, when it's actually something deeper and more mysterious and independent of ourselves. The conviction of the Holy Spirit is His loudest voice, I believe.
The best way to describe it would be our own reaction to sin. Even if we engage in a sinful behavior, we hate it and possibly hate ourselves, too, afterwards. This simply isn't the case for the vast majority of unbelievers. If they get away with sin and can continue on with life as normal, they're convinced that they're the smartest people alive and feel great about themselves. Engaging in sin and getting away with it is the same kind of victory that conquering a sin and putting it away for good is for a Christian.
