believe：相信某事物是真实的（或者相信某人所说的话）

If you believe something, you think it is true.

如果你相信某事物，那么你认为它是真实的。

If you believe someone, you think they are telling the truth :

如果你相信某人，你认为他们讲的是实情。

Nobody believed what I said.

没有人相信我所说的话。

I didn't do it - you've got to believe me!

我真没有做那件事情，你应该相信我（相信我所说的话）。

永远相信自己能行。

believe in

1. You think something exists : 你认为存在某事物。（trust in)

Do you believe in God?

你相信这个世界上有上帝吗？

Do you believe in ghosts？

你相信这个世界上有鬼吗？

2. You think that something is good : 你认为某事物是好的。

He doesn't believe in private education.他不相信民办教育。

3. You think someone is good and will achieve good things: 你认为某人很棒，将来会很有出息。

My mother always believed in me.

我妈妈始终信任我。

相信自己很棒，相信自己能行。

用一句话记住他们的区别：

The crazy man said: "do you believe in ghosts? I saw a ghost in a forest last night. I told many people about it. But nobody believes what I said."

那个疯子说道：“你相信这个世界上有鬼吗？昨天晚上我在森林里看到了一个鬼。我告诉了很多人，但是没人相信我所说的话。”