Actually, I think it is just about that simple. We just follow along in the Bible and take note when things change. Like when the Law was given. Or when the New Covenant began. When Israel rejected the Gospel, and it went to the Gentiles.The term seems to have taken on a meaning that I am sure I don't even know.
My first idea was that it meant "first the Jew then the Greek" (first Israel proper and then all the other people). Which I thought...Well, duh, two folds of Christ, two dispensations, of course! Too bad it's not that simple, it should be. But apparently there is more religiosity and the teachings of men to it than that.
Good luck!
Abraham believed, he would have lots of children. God's Word.
The disciples believed, Jesus was the Annointed King of Israel. Also God's Word, but a different Word.
We believe, if we confess with our mouth the Lord Jesus, believing God raised Him from the dead, and we will be saved. Also God's Word, but a different word.
All are saved by faith in God's Word, but the Word given was a little different, depending on the time, on the degree of revelation, and the people God was addressing.
For @ByGrace 's benefit, hyperdispensationalism would be where we would say, we are the New Testament Church, so we only read Paul's letters, for example.
Much love!