Believe can mean a whole-hearted (no doubt) trust in something that it is true/certain, or it can mean thinking something to be true/certain. In other words, believe can be interchanged with "trust" or "think". "Trust" cannot be interchanged with "think". Here are examples: "The family believed in the fireman to save them from their burning house." "The family trusted in the fireman to save them from their burning house." "The family believed the candle would not start a fire." "The family thought the candle would not start a fire."

Victor explained it well (Belief is for a single action, Trust is for the person/object). Trust means reliability. Belief means you accept it as true. "I've worked with Bob for years. I trust Bob to finish the job on time." In this example because there is a history of reliability you use trust. "John was highly recommended by Dave. I believe John will finish the job on time." In this example you don't know John well enough to rely on him but you can still think he's able. If John does a good job then next time you may trust him. "I trust Dave's recommendation" The trust is in the recommendation. The trust isn't in Dave. "Fred was late last time but I believe he will finish this job on time" Fred can't be trusted in general.

Believe and trust can sometimes be similar But I think, believe -- is usually about a specific statement, trust -- is about a person's character. A: I didn't steal your money B1: I believe you (you didn't steal the money) B2: I trust you (you aren't the type of person who would steal money)