They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them (ESV)
We often hear that we are all born in God's image, but we struggle to fully understand what that means. The Apostle Paul helps us to understand one concept of being born in God's image by telling us about our conscience. Each person is born with some level of conscience, that is, an innate sense of right and wrong.
Our conscience is God's law, at least natural law, that is written on the hearts of all mankind. The very first human child, Cain, experienced the work of conscience after he had murdered his brother Abel. He cries out, "Am I my brother's keeper?" (Genesis 4:9). Indeed, he was his brother's keeper. He was responsible for breaking God's law, which was written on his heart.
Atheists can claim that there is no God, yet this nagging thing we call conscience speaks to God's existence. If we were simply creatures with no purpose, who arrived at this earth accidentally, with no judgment to come, then why would we have a built in conscience that would accuse or excuse us? This internal code of law speaks of a greater law giver. We know him as our Creator.
It is important to note that while our conscience can point us to a Law giver it cannot save us. Even the Old Testament Law that was given specifically to the people of Israel lacked the ability to save even one of its followers. The Law is able to condemn us. It perfectly points out where we have failed. Only God can redeem those who have broken his law.
As you speak to people who claim that there is no God ask them if they feel any remorse for anything they do (lying, stealing, dishonesty, etc.). Point them to the source of the internal law that condemns them.