Here's how I've come to view it, I have kind of a middle-ground position.
1. Christ died, in a collective sense, for the whole world and all people are invited and commanded to enter into the New Covenant by repentance and faith.
2. Christ also died specifically, in a particular sense, for all who would ever believe in him. For these people he actually bore their deserved judgement and removed their sins in his blood.
3. All people are able to enter into the New Covenant, meaning they are able to repent and believe, but no people are willing to. People are not willing because they hate God and love sin, so they choose, without exception, to reject his offer.
4. In God's grace and mercy he has chosen some to bring into this New Covenant by overpowering their unwillingness with his sovereign grace, and he makes this happen by regeneration, a supernatural work of the Holy Spirit whereby a sinner is brought from spiritual death to spiritual life, the outward sign of this supernatural work being true repentance and saving faith.
I hold this position because the bible affirms a few things very clearly. It affirms unconditonal election from cover to cover. It also affirms that man is totally depraved and spiritually dead. It also differentiates between the general call of the Gospel and the effectual inward calling of the elect. It further affirms the eternal security, or certain perseverance, of the saints. However, it also does affirm that the atonement was both particular and universal.
Hence, I'm basically a four-point Calvinist. Five-point Calvinism may be more logically comfortable for our fallible human minds, but the bible seems incredibly clear on the simultaneously universal and particular nature of Christ's atonement.