Now faith is the substance of things hoped for, the evidence of things not seen. - Hebrews 11:1
I have often heard this verse used to define faith as believing in something without evidence. In fact, the NIV rendering has that feel. However, I don't really believe that is what the verse is pointing to.
If you look at the context, the author of Hebrews isn't discussing what faith is, but what faith does. It is by their faithfulness to God that God is pleased by the saints of the OT. It is by faith that those things that we hope for comes to pass.
The word translated as 'substance' above is the Greek word 'hypostasis', which literally means substance. It refers to the fundamental aspect of something; its core, its foundation, its essence. Thus faith makes those things which we hope for manifest. It brings them into reality.
The word translated as 'evidence' is the Greek word 'elegchos' which is derived from 'elegcho' which means "to convict or try". Thus 'elegchos' is a legal term referring to those proofs by which a conviction is made. It can also refer to the conviction itself. However, the meaning "conviction" is unlikely considering that when it is used in this sense it is meant in a condemning sense. However, there is nothing condemning about faith. Therefore it is far more likely that "proof" or "evidence" is more likely. So what does it mean to be "the evidence of things unseen"? It means it is by faith that the unseen things are tested for validity.
If you think about it, all of this makes sense. Faith is essentially trust. You can have faith in seen things and unseen things, but irregardless of what it is you have faith in, your faith is the basis upon which you act. I sit on a chair because I trust that chair to support my weight. I married my wife, in part, because I had faith that she would be true to me.
Therefore, when I have faith in something hoped for or unseen, I act on that faith. If what I hope for is valid, then my actions will bring it about. When that happens, it becomes evidence for acting on it in the future. Indeed, if we look at the rest of Hebrews 11, we do not see saints who hold onto their faith despite evidence, such as Job or Joseph (in the prison). Instead we see men who acted based on their faith in what God had told them and who changed the world through those actions.
This is essentially a call for anecdotal evidence. Paul is calling us to go out and live out our faith for the sake of collecting that anecdotal evidence for the gospel. You want to convince people of the truth of God? Go out and live it. When you do, you will receive confirmation after confirmation of God working in your life that you can then use to tell people of the faithfulness of God. Each day, each moment we have an opportunity to demonstrate the wonders of God if we choose to live His way.
So go out and spread the word by living out the word. As James said, show people your faith by what your faith can do.