Firstly, to clarify, atheists believe that there is no god, as opposed to theism (which is a belief that there is a god or gods). As Christians, we have a monotheistic view (one God) existing in a Trinity (the Christian belief of God existing as the Father, Son, and Holy Spirit – 3 in 1, one nature, three persons).
The reason for this question being so vague is that there are many various reasons why people don’t believe in God (or a god, for the matter). The most common major objections are: the problem of evil (there is evil in the world, so there can’t be an all good and loving God); the Big Bang/Evolution vs. Creation debate; God didn’t answer my prayers; and I’m doing fine, so why do I need God?
The main approach to evangelizing to someone who doesn't believe there's a god is to ask or find out: Why he/she doesn’t believe that there is a god? This is the key clincher question to approach this topic. We have to know where they are coming from, and ultimately believe and trust that God will meet them where they are at. This is our hope and confidence.
Here's what we can do:
- pray and ask God to reveal Himself to that person
- recognize that it is God, through the power of the Holy Spirit who moves and changes a person’s heart and not our own efforts
- pray for strength, discernment, and courage to witness to this person and for God to speak through you
- devote yourself to following Jesus that others may see Him through you
- ask yourself, “why do I believe God exists?” and share your personal answer with them
- Come alongside them, get to know them, see what they are all about, befriend them, show God’s love to them
[Answered by Jon]