TULIP is an acronym used to summarize the five points of Calvinism. They are laid forth clearly in the Canons of Dordt. They are set down in response to the teachings of Jacob Arminius. After his death, his followers put forth the Remonstrance, putting forth their views on five points of disagreement with the prevailing notions of Christian theology at the time.
With this in mind, we can make a few qualifications. These five points do not summarize all that the Bible teaches upon the sovereignty of God, but rather put forth doctrine on points of disagreement. TULIP is a convenient acronym, but not necessarily the suggested mode of presenting Calvinism. The term Calvinism is misleading in that it is put forth as doctrines that John Calvin taught. While this is true, many (if not all) Calvinists would say they believe in Calvinism because they find it to be biblical, not because of commitment to a person. I presume “Arminians” would say the same.
The points and a quick summary of what they are as follows. I'll tie them together logically at the end.
T – Total Depravity
Man is, by nature after the fall, sinful. Not only have they committed sins, but the sinfulness (depravity) is total, extending to every single part of their being. This is not to mean that they are all mass murders, but simply that they cannot, without the grace of God, will anything that is good or pleasing to God. Even those works that look “good” to us, are not done in faith and thus are not pleasing to God. There is nothing good in man after the fall.
U – Unconditional Election
God's choice to save people is unconditional, based upon His will and His will alone, and not upon anything that they have done, earned, or will do in the future. Notably, His election of people is not based upon his foreknowledge of their decision to choose Him. Their faith is a gift of God as well, and dependent upon His election of them.
L – Limited Atonement
Christ's death secures salvation for the elect (those that God has chosen). Those that He died for have their sins forgiven. The word “Limited” refers to the extent of the atonement (Christ's death). If Christ's death secures salvation, then the fact that someone isn't saved means that Christ didn't die for them. (Limited carries a negative connotation, which is unfortunate)
I – Irresistible Grace
For those that God has chosen, there comes a point where God will save them by the working of His Holy Spirit in their lives, bringing them to repentance and faith. This call, done by God's grace, is thus termed “irresistible” as He overcomes all obstacles to faith and saves people.
P – Preservation of the Saints
For those God has chosen, He will also preserve them to the end, so that they will not fall away (they will remain saints).
These five points (rough summaries), are set in contrast to the five points brought up by the followers of Arminius, which were: Partial depravity, election based on foreseen faith, universal atonement, resistible grace, and the possibility of falling away from faith.
They tie together nicely as follows:
If someone is totally depraved and unwilling to choose good, then, on their own, they cannot and will not choose God. God's choice of saving them cannot then be based upon anything they do (since they do nothing good), but upon His will alone.
If someone is unconditionally chosen by God, then that means God much do the entire work of salvation. Namely, Jesus' death upon the cross must be sufficient to save them. (Total Depravity again). This means His death is the sufficient and necessary condition for salvation, and Christ did not die for everyone (as, if He did, then everyone would be saved).
If Christ's death is sufficient to save someone completely, God has unconditionally chosen them for salvation, and they, on their own, are totally depraved, then that means God must break through their hardness of heart to bring them to faith and repentance. Left to their own devices, they would resist God until the bitter end. The only way they're going to be saved is if God completely transforms their heart, so that His grace is then irresistible.
If God has unconditionally chosen to save the elect and He has provided all things necessary for their salvation in Christ, then as part of the provision in Christ, He has also provided an endurance to the end in Christ.
Notice that if we deny one point, we will eventually deny all points.
As an example:
Grace is resistible -> Man is not totally depraved (for if they were, they'd resist God's grace to the point of death, but some are saved) -> God's election not based completely on self (resisting or not is also a determining factor) -> Those that are saved can fall away (As if they can resist now, what stops them from resisting later?)-> Christ's death isn't sufficient
One point also implies the others.
People are preserved to the end - > Since this must be effected supernaturally, it must be conditioned upon God and not man (U) -> Christ's death is sufficient to save (L) -> Man cannot add to his own salvation (T) -> God, applying Christ's death, cannot be resisted. (I)