The Greek word is καταργέω, which means to destroy, nullify, or render impotent. The word occurs 26 times in Christian New Testament: please click here to view each respective context, usage, and translation; and three times in the Greek translation of the Hebrew Bible, which is the Septuagint: please click here to view each respective context, usage, and translation.
In the Book of Romans, the Apostle Paul develops an argument that faith ALWAYS has been the basis of right relationship with the Lord. (Please click here for other relevant background info in this regard.) That is, the Law of Moses was about "do this, and this will happen" or "don't do this and this will happen," and so was not based on faith (Rom 10:5 and Gal 3:12). However, behind the Law of Moses there was the "requirement" of the law, which was to love the Lord with all of ones heart and soul.
Deuteronomy 10:11-13 (NASB)
11 Then the Lord said to me, ‘Arise, proceed on your journey ahead of the people, that they may go in and possess the land which I swore to their fathers to give them.’
12 “Now, Israel, what does the Lord your God require from you, but to fear the Lord your God, to walk in all His ways and love Him, and to serve the Lord your God with all your heart and with all your soul, 13 and to keep the Lord’s commandments and His statutes which I am commanding you today for your good?
The promises mentioned in verse 11, above, were the basis of faith (that is, the Promised Land stemmed from the Abrahamic Covenant). Therefore what changed was verse 13: that is, the New Covenant, which is an extension of the Abrahamic Covenant, had replaced the Mosaic Law. (Jesus inaugurated the New Covenant, which replaced the Mosaic Law.) What has not changed was the "requirement" of the Law, which was to love the Lord with all your heart and with all your soul (mentioned in verse 12, above); thus what has changed was just the Law of Moses (mentioned in verse 13, above).
Another relevant passage comes from the prophet Micah, which makes explicit inclusion of the love of ones neighbor as part of this "requirement."
Micah 6:7-9 (NASB)
7 Does the Lord take delight in thousands of rams,
In ten thousand rivers of oil?
Shall I present my firstborn for my rebellious acts,
The fruit of my body for the sin of my soul?
8 He has told you, O man, what is good;
And what does the Lord require of you
But to do justice, to love kindness,
And to walk humbly with your God?
These verses from Micah are what tied the "requirement" of the Law to include not just the love of the Lord, but also the love of ones neighbor as oneself (that is, what is implied is that "justice" and "kindness" are seen not just in ones relationship with the Lord, but with ones fellow human beings). Thus Jesus indicated that these two commandments were the overriding, most important laws: loving the Lord and loving ones neighbor as oneself (Matt 22:26-40). In other words, Jesus's inductive reasoning to tie the concepts together was very consistent with the Law of Moses, because of the following verse.
Leviticus 19:18 (NASB)
18 You shall not take vengeance, nor bear any grudge against the sons of your people, but you shall love your neighbor as yourself; I am the Lord.
So when the Apostle Paul states that we are not nullifying the Law through faith (νόμον οὖν καταργοῦμεν διὰ τῆς πίστεως...), he adds that we are fulfilling the "REQUIREMENT" of the Law, which is to love the Lord with all our heart, and to love our neighbor as ourselves. Thus we "fulfill" the ("requirement" of the) Law, which is echoed in the New Testament (please see Romans 13:8-10; Galatians 5:14; and James 2:8).
What has changed is the role of the Mosaic Law, which was replaced by the New Covenant. The New Covenant, though, does not change the "requirement" of the Law, which was to love the Lord with all of ones heart and soul, and to love ones neighbor as oneself. (This "requirement" also appears to have been the "proto"-Law on earth before the giving of the Law of Moses on Sinai--please click here.) So Paul ends the third chapter of Romans with the words: On the contrary, we establish the Law. To put it another way: it is not the substance that has changed (i.e., "requirement" of the Law), but instead the form has changed (i.e., that Mosaic Law was replaced by the New Covenant as the primary vehicle for approaching the Lord). In either case, faith ALWAYS has been the basis of right relationship with the Lord.
In conclusion, the form has changed.... however, believing by faith has not changed, and the "requirement" of the Law has not changed; in these latter two respects therefore we establish the Law.