What does Piper mean when he says he's a seven-point Calvinist?
When John Piper says he is a "seven point Calvinist," he does so half jokingly and half seriously. Historically, there are five points of Calvinism, not seven. Piper isn't seeking to add two more points, but is simply calling attention to his belief in the traditional five points (total depravity, unconditional election, limited atonement, irresistible grace, and perseverance of the saints) in a way that also points toward two additional "Calvinistic" truths that follow from them: double predestination and the best-of-all-possible worlds.
The "sixth" point, double predestination, is simply the flip side of unconditional election. Just as God chooses whom He will save without regard to any distinctives in the person (Ephesians 1:5-6; Acts 13:48; Revelation 17:8), so also he decides whom He will not save without regard to any distinctives in the individual (John 10:26; 12:37-40; Romans 9:11-18; 1 Peter 2:7-8). By definition, the decision to elect some individuals to salvation necessarily implies the decision not to save those that were not chosen. God ordains not only that some will be rescued from his judgment, but that others will undergo that judgment. This does not mean that someone might really want to be saved but then be rejected because they are on the wrong list. Rather, we are all dead in sin and unwilling to seek God on our own. A true, genuine desire for salvation in Christ is in fact a mark of election, and therefore none who truly come to Christ for salvation will be turned away (John 6:37-40).
So just as God doesn't choose to save certain people because they are better than others (unconditional election), neither does he choose not to save certain people because they are worse than others (unconditional reprobation, or double predestination). Rather, everybody is lost in sin and no one has anything to recommend them to God above anyone else. And so from this mass of fallen humanity, God chooses to redeem some and leave others.
The "seventh" point, the best-of-all-possible worlds, means that God governs the course of history so that, in the long run, His glory will be more fully displayed and His people more fully satisfied than would have been the case in any other world. If we look only at the way things are now in the present era of this fallen world, this is not the best-of-all-possible worlds. But if we look at the whole course of history, from creation to redemption to eternity and beyond, and see the entirety of God's plan, it is the best-of-all-possible plans and leads to the best-of-all-possible eternities. And therefore this universe (and the events that happen in it from creation into eternity, taken as a whole) is the best-of-all-possible-worlds.
More on Calvinism & the Doctrines of Grace.
John Piper, The Justification of God, chapters 5, 9, 10, 11.
Wayne Grudem, Systematic Theology (Zondervan, 1994), chapter 32, "Election and Reprobation."
Jonathan Edwards, Concerning the Divine Decrees in General and Election in Particular, in The Works of Jonathan Edwards Volume II (Banner of Truth, 1974), 525-543.