Because is used when the reason is the most important part of the sentence or utterance. The because clause usually comes at the end: 'I went to Spain last summer because I wanted the guarantee of sunshine on every day of my holiday.'
As and since are used when the reason is already well known and is therefore usually less important. The as or since clause is usually placed at the beginning of the sentence: 'As the performance had already started, we went up to the balcony and occupied some empty seats there.' 'Since John had already eaten, I made do with a sandwich.'