For the word of God is living and active. Sharper than any double-edged sword, it penetrates even to dividing soul and spirit, joints and marrow; it judges the thoughts and attitudes of the heart. - Hebrews 4:12
God is real, and alive. Convincing the skeptic of this is not something that you or I can do (unless it's you reading this, God.)
What I can do is describe what I believe, and maybe if I have time and inclination, why I believe it. In a sense it's nontrivial and in a sense it's so simple that it's unbelievable.
The central tenet of Christianity is God. I, as a Christian, believe that God reveals certain attributes about himself to us. God is real, and alive. God is all-knowing and all-seeing. This means that he knows, using the philosophical definition of "knows," everything that has ever occurred, everything that is currently occurring, and everything that will ever occur. God is not only infinitely good, but he is the definition of good. Evil, which is done by angels and men alike, is recognizable because it contrasts with the nature of God. Since God is infinitely good, he is infinitely just, which is why he does not prevent people from doing bad things. This is also why he holds us accountable for the bad things that we do.
This accountability is a big hangup that some people have with God. His justice demands that wrongness, or not-like-God-ness, not be condoned. There is a penalty for sin that must be met. The penalty for being unlike God is to be apart from him. This is what some people call hell: the state of being apart from God. In the Bible it's compared to a giant, burning garbage dump. All humans are unlike God because we all do wrong things, at least some of the time. True justice demands that we face the penalty for our sins.
Fortunately for us, God is also infinitely merciful. He created a way for us to bridge the gulf between humans and himself. He himself came to earth in the person of Jesus Christ and died in order to soothe the demands of justice. He committed no sins, but took responsibility for all of the sins committed by all of the people, past and future. He then rose from the dead, demonstrating his power over life and death.
God offers us free of charge the benefits of his sacrifice. He literally gives us grace, which is unmerited favor: a gift that we do not deserve and can never earn ourselves. In order to receive this gift, we simply have to believe, and trust God. When this happens, God changes us and we begin to become more like him. Someday, after we die, our conversion from sinful people to sinless people will be complete and we will be able to be in the presence of God. This is what some people call "heaven."
That's the nuts and bolts of it. There's all kinds of other stuff that people have added on, some of which is correct and some of which is not. But this is the important part.
For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in creation, will be able to separate us from the love of God that is in Christ Jesus our Lord. - Romans 8:38-39