Trust is important for relationships. I think any relationship is built on trust. But the kind and amount of trust depends on the relationship.
I go to a bar and a guy at the door asks for money. I trust that he is legitimately collecting a cover charge and give him money. Trust. I don't ask him to proove he is legit. (And those collecting cover charges don't wear uniforms or nametags to identify themselves.)
I go to the grocery store and buy many items. I trust that the items will be rung up at the correct prices, and that the total will be correct.
I pay cash for something I buy. I give a $20 bill for a $5 purchase. I trust that the person will give me $15 back.
Trust is a part of friendship too. Though not so neatly defined as in customer / store relationships. Friendships vary. And with that variation, comes variation in the level of trust.
We feel safe in a relationship when the level and type of trust fits the relationship. I feel comfortable with the clerk at the store if I can expect her to ring up my purchase correctly and give me correct change. As well as trusting her to be reasonably civil and to not jump out and attack me or any such.
I think much of the hurt in relationships comes from either broken trust, or expecting to have a relationship with a higher trust level than what one has.
Self-trust is a factor too. Can I trust myself to deal with what the other person might do? The more I trust myself, the safer I feel. Feeling safe comes from a combination of trust in the other (or the general environment) and trust in oneself.
Sometimes, when someone has broken our trust, it makes it hard to trust others, hard to trust new people in our lives. I think the first step in learning to trust again is to build that self-trust. If I can trust myself to deal with what might happen, then I can take a chance on a relationship.