(Credit for this mostly to R' Menachem Leibtag)
The core of the slichot (and Yom Kippur) prayers is the 13 Middot or Attributes of God. In selichot, they are recited over and over with various songs and poems in between. At first glance they seem like a kind of magic spell which instantly gets you forgiveness for your sins. But that would be very strange and, I think, foreign to Judaism. What, in fact, do they mean? Why are they recited?
If you look at the Ten Commandments, you will notice something interesting. They include not just commandments and reasons for the commandments, but also descriptions of God. For example, God is "zealous" ("kana", 2nd commandment) and unforgiving to sinners (3rd commandment).
Historically, the Ten Commandments were written on tablets and given to Moshe. But after the sin of the golden calf, Moshe broke the tablets. This symbolized a more important fact: that the covenant signified by the 10 Commandments was now nullified. It had to be, because a zealous and unforgiving God could not coexist with a sinning Jewish people. When Moshe went back to God, there were two options. Either the Jewish people would be annihilated. Or else, God's attributes would have to change.
If you make a list of all the attributes of God in the second and third Commandments, and compare them to the 13 Middot (though the order is different), you will find that the "zealous" attributes in the Commandments have "non-zealous" parallels among the Middot. For example, "el kana" is replaced by "el rachum vechanum", "lo yenakeh" by "venakeh", "poked avon" by "nose avon", and so on. After the sin of the golden calf, God made a new covenant with the Jewish people. The actual commandments were unchanged (and so did not have to be repeated). But God's manner of relating to the Jewish people did change. Absolute justice was replaced by a mixture of justice and forgiveness. The changes were recorded for posterity in the 13 Middot, which represent a new set of tablets and a new covenant.
Obviously, God cannot forgive every sin, all the time, nor would that be desirable. After the 13 Middot, the Torah still sometimes mentions God's zeal, jealousy, and unforgiving justice. Sometimes God will be merciful, and sometimes not. Whenever we mention the 13 Middot in prayers, we are alluding to both covenants. We are praying that God act according to the later, merciful covenant, though the earlier model is still an option as well.
"Zechor lanu hayom brit shlosh esreh" - "Remember for us, today, the covenant of the 13 attributes"