Our Father which art in heaven
What is a father? The bible definitely has a lot of references to God as our Father, and because I have a father and I am a father (no offense to those who aren't, and by no means am I an authority), I think it means a few things: we depend on Him, we should have a relationship with Him, and we should want His guidance.
What is heaven? I like to think of this physical universe as an ever-expanding bubble that has laws that govern things like time (i.e. always linear) and space (i.e. x-y-z axis where no two objects occupy the same coordinates), and then there's a spiritual dimension that's everywhere: outside of that aforementioned bubble; heaven would then be a protective bubble (inside the spiritual dimension), which God protects (and then hell would be the unprotected areas).
Hallowed be Thy name
We just shouldn't treat Him lightly; He should be treated with respect. One of his names is I Am (see Exodus 3:14): he upholds everything (see Hebrews 1:3), so if He ceased to exist, everything would cease to exist; don't mess with that power, honor it!
Thy kingdom come, Thy will be done in earth, as it is
in heaven
What is God's will? One thing for sure, He wants to restore our physical universe to the way He intended it to be; but, while we wait for that, we must try to understand and accept God's temporal plan for us as parts in the same body (see Matthew 5:29-30, Romans 6:13, Romans 12:4-5, 1 Corinthians 12:12-27).
Give us this day our daily bread.
Who provides for you? Some people think they provide for themselves, but it's God! (Psalms 37:25, Philimon 4:19).
And forgive us our debts, as we forgive our
debtors.
How can we expect Him to forgive us if we don't forgive others? In
areas you've been offended: Don't curse it, don't nurse it, don't
rehearse it, don't disperse it, and God will reverse it (you like that?!).
1 John 1:9-10, Psalm 139:23, 2 Corinthians 7:1,10, Ephesians 4:30-32,
Matthew 18:22, Matthew 6:14-15
And lead us not into temptation,
God is good and won't put us in situations we can't handle; and,
when He does, we need to realize that it is like the hammer to iron: a
process that seems painful, but forges us into a better utensil (see Proverbs 27:17).
but deliver us from evil
James 4:7, Romans 13:14, Ephesians 6:11-18