Evangelical Christians throw the phrases "believe in Jesus" and the more-or-less synonymous "ask Jesus into your heart" around freely, so much so that they've entered into American pop religion. The problem is we rarely take the time to explain what they really mean which can result in people having a very confused, un-biblical understanding.
So what does it really mean to "believe in Jesus?"
What it isn't
First, let's look at some things that are commonly mistaken for the real thing. Believing in Jesus isn't just believing that he's real or that he really was a historical person. It's not even believing that Jesus really did rise from the dead.
It's not believing Jesus cares about you. It's not praying to him. It's not believing he will heal you or help you with your problems with work, money, or family.
What it is
The New Testament word we translate as "believe" carries a lot of meaning with it. It includes trust, reliance, and dependence on the object of the belief.
An example would be believing a chair can hold you up. Standing there looking at it, saying it could hold you up isn't biblical belief. Sitting on the edge of the chair so that some of your weight is still on your feet isn't biblical belief. Sitting solidly in the chair with you feet in the air, so that you would fall if it broke, is believing the chair will hold you up.
How does this relate to Jesus? To believe in Jesus, you have to
1) agree that you need Jesus to forgive your sins
2) believe that his death was sufficient for the forgiveness of your sins
3) give up any notion of ever being good enough to please God/go to heaven/earn forgiveness on your own.
Additionally, Jesus always talked about belief paired with repentance. You aren't called to "believe" but "repent and believe." So along with the above, you also must decide to change the way you live — from doing it however you feel is best to living the way Jesus says.
So to "believe in Jesus" you have to trust solely in Jesus' work on the cross for the forgiveness of your sins and devote your life to following Jesus. It's a life with no other safety net; if Jesus fails you, you're doomed. But that's OK. Jesus won't fail.