In the phrase "faith is the substance of things hoped for," Paul is not really defining what faith is, but rather he is showing what faith does in an operative sense: Faith undergirds what we hope for. Substance means "that which stands under." Faith is the foundation for what we hope, the foundation for our relationship with God and everything that it implies within His purpose. Faith is the very beginning of everything that really matters spiritually.
By saying that it is the "evidence" or "assurance" (the word can literally be translated "title deed," but "assurance" seems to be the best all-around word) of things hoped for, the author comes much closer to defining what faith is. In its simplest form, faith is merely belief. As our understanding becomes more complex and operative, when we begin to put faith to work, it becomes "confidence," and finally, in its best form, when it becomes fully operational, it is "trust." This trust, this full measure of faith, is alive and works within our relationship with God.