Common parlance in the Mormon Church is to say we "know" things when we bear our testimonies. We know the church is true. We know that God lives. We know there is a prophet on the earth today.
For some, that declaration of knowledge is disquieting. For many people, we only can know things we perceive with our senses. Or things that are supported by empirical evidence. I had a friend a number of years ago who made what for me was a startling statement. He was questioning his own activity in the church, and he declared first that he could only accept things that could be demonstrated empirically, and then he made the astounding claim that he believed that all the great scientific discoveries had all already been made. This was about 20 years ago. Seems to me that he was at least wrong about all the major scientific discoveries having already been made by the late 1980's.
When we way we know in the church, our use of the word knowledge links it closely to faith. In Alma 32, we learn about the classic example of faith – plant the seed of desire, nurture it and if it grows, then we have faith, which ultimately leads to knowledge. Another example I've heard is this: we "know" when we push that switch on the wall that the lights will come on. Now, it's not actually true that every time we push the switch the light comes on; sometimes the power is out, or the circuit breaker has been tripped or the light bulb has burned out or the switch breaks. But still, we "know" because of our long experience with wall switches, that flipping the switch means the light should come on, so we say we know it will happen.
To be sure there are some who do know. Joseph Smith saw God the Father and Jesus Christ, so he knew they existed. Those who find him a credible witness might also say they know because they trust him. Joseph and Sidney Rigdon also had a separate experience described in the Doctrine and Covenants in which they report having seen the Father and the Son. Stephen in the New Testament makes a similar claim. Just as a jury knows by the witnesses presented at a trial, one way we may know is to listen to credible witnesses, even if we don't have first hand knowledge.
Another way we know may be our own cause and effect relationships. We keep a commandment and we perceive a blessing in our lives as a result. One might argue coincidence in such an example, and one might similarly wonder why one person sees the blessing in his life and another apparently does not. But for the person who perceives it, the cause-and-effect relationship is real.
Perhaps the greatest way of perceiving spiritual truth, however, is in the change that we feel in our hearts. Book of Mormon prophet Alma teaches that at our conversion our hearts change. He's not the only one; the Old Testament speaks of Saul's receiving a new heart when he becomes king, for instance. Alma also speaks of feeling the swelling of the spirit in our hearts. The Doctrine and Covenants also records the idea that we might understand the spirit through our heart and mind.
For me, I have to work hard to separate my own emotional response (I cry at most movies, and even have an emotional response to favorite classical music, and I'm a sucker for family-oriented advertising) from what I think is a spiritual response. For me, a spiritual response is not always an emotional one (though for others it may be). But over time, I have come to sort out what a spiritual response looks like to me. Having that response allows me to say "I know" even if my knowledge is not empirical. And I suppose that for many, my "knowledge" is only an expression of my faith. That's ok for me.
I'm struck by the many years in the Book of Mormon that prophets spoke so clearly about the coming of Christ long before He came, and spoke as if He had already come. These prophets spoke based on their own prophetic knowledge, and their people often acted in faith as if it had already happened. When we speak in faith of "knowing", I believe it is a similar circumstance.