Having Faith in something is believing something is true (or false) without it being Proved to be true (or false). Note... this does NOT mean that one cannot have Evidence for the thing being true (and still call it Faith)... only that it has NOT been Proved to them.
Note... some people use the word Faith as a synonym for Worship. However, this is incorrect. Faith means to believe in (or disbelieve in) something without proof.
Faith and God
For example, a person can choose to believe in God, choose to NOT believe in God, or not even make a choice. (A person who has not made a choice may have NOT even pondered the question.)
A person who chooses to believe that God exists has Faith in that. (Examples of these people include, but are not limited to, believer of Zoroastrianism, Judaism, Sabianism, Christianity, and Islam.)
A person who chooses to believe that God does not exist has Faith in that. (These people are sometimes called Atheists; however Atheists either disbelieve in God or have NOT decided to believe or disbelieve in God... which makes usage of the term "Atheist" somewhat confusing.)
(A person who has not choosen to believe either does NOT have Faith in either.)
Faith and Science
Faith also comes in other things. For example, Faith exists in Science too. (Despite many people who call themselves Scientists saying it doesn't.)
Mathematicans can tell you that to have Proved something for something infinite (like an Infinite Series), you MUST prove it for all elements of the series... and not just some of them. Showing something is true for only some of the series is NOT proof. However, in Science we do just this.
In Science you often gather Evidence for something only a finite number of times. After all, doing an experiment 10 or 100 or 1000 times is still NOT all the time. (I.e., Time is infinite, thus to prove something we MUST do it for all the time.)
For example, Ohm's Law talks about electricy all the time. But this is based on experiments only done a finite number of times. And even though it continues to be true every time we do the experiment (as far as I know), still that is NOT Proof. Theorectically, we could get different results in the future when performing the same experiment.
The thing is, many people believe this to be true! In other words, they have Faith in this.
If people just thought that it is reasonable to do engineering as if Ohm's Law was true (but not actually believing or disbelieving it), then it would NOT be Faith. But many people actually believe it is true, thus making it Faith.
In Science you do NOT prove things. You show "probablies". This does NOT make Science useless. In fact, the Scientific Process is extremely useful! However, one should still be mindful of the true nature of Science and that, for many, it too is based on Faith.
Faith and Axioms
By definition, Axioms are used as if they are true. If one actually believes that an axiom is true, then they have Faith in that.
Faith and Premises
Like Axioms, by definition Premises are used as if they are true. If one actually believes that a premise is true, then they have Faith in that.