"Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord; And He shall send Jesus Christ, which before was preached unto you:" (Acts 3:19)
That was Peter speaking to a crowd in Jerusalem. Jesus preached a similar message: "From that time Jesus began to preach, and to say, Repent: for the kingdom of Heaven is at hand." (Matthew 4:17, emphasis mine) The "kingdom of Heaven" refers to the fact that now, God can dwell in our hearts by the Holy Spirit. He can set up shop within and live inside of us never to depart.
Looking at the first verse, Peter places a qualifier on the "times of refreshing [that] shall come". Notice, it's after we repent. Paul, speaking of repentance in his second letter to Timothy, says that "God peradventure (perhaps) will give them repentance to the acknowledging of the truth."
Taking these three verses out of context (admittedly) and stringing them together, you get a sequence that goes something like this:
Acknowledging the truth. Whether it be about a situation, about our role in what went wrong, or simply about Jesus and Him being the truth. Acknowledge. From there, to Repentance. Notice it says that God gives repentance. Repentance, much like faith, is something that's given. God wants everyone to come to Him. What enables us to become right with Him, however, is a gift, plain and simple. "That no flesh should glory in His presence." (1 Corinthians 1:29)
And after that? What comes after we repent? Is it a stillness that envelopes us like a vacuum of holiness? Must've been what the Garden of Eden was like before the fall. It's like God restores us to a level of peace and contentment. But it doesn't mean that we'll be without the temptation to sin again. But don't worry about it. The times of refreshing are coming from the Lord. Look for them.
Staying the courses
Watch out though. I still don't quite understand this next passage. It's somewhat disconcerting, not because I still feel like sinning at times, even after I've repented, but because upon a cursory reading, it looks like my salvation might be negated with the slightest slip-up:
"For if we sin willfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins, But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries." (Hebrews 10:27) I'm going to leave this in here, only because it's something that must be worked through by every believer who believes in the sacrifice of Jesus to make us right with God. Wrestle with it, but not with God.
And so, moving along (how can one move along from a passage like that?), it says "when the times of refreshing shall come from the presence of the Lord". I do recognize times in my life where there is that stillness that precludes a sudden rush of blessing from the Lord. And while all of this is necessarily referring to those times, wonderful as they are, it's more important to know that they too shall pass. Because it isn't about the spikes of beauty that we're sort-of holding our breath for, it's about walking with Him day by day.