Dispensationalism is a very common and widely accepted theological view of Biblical interpretation. This viewpoint basically says that there are different "dispensations" (distinct periods of history) in the Bible where God deals with people in different ways.
At this basic level, it's hard to have much disagreement with that definition seeing that certainly God dealt differently with the Old Testament nation of Israel, compared with the New Testament believer. However, dispensationalism often takes it too far by creating solid barriers within the Biblical text where they shouldn't exist.
In contrast with replacement theology, dispensationalism sees a clear distinction between the church and ethnic Israel, and generally believes that only Israel will go through the 7-year tribulation period at the end of the age.
Some dispensationalists assert that people were saved in different ways during different periods of history. However, the Bible makes it clear that people have always been saved by faith.
There's a sect called "hyper-dispensationalism" that claims there are 15 (maybe more?) different dispensations in the Bible! This is a bit absurd.
It's easy to be committed to a particular theology, but often people then take that theology and interpret the Bible based off their theology rather than let the Bible speak for itself. We cannot force-fit God's Word into our man-made concepts.
I'm not anti-dispensationalism, but it really doesn't do that much for me. It's a framework of Biblical interpretation, but its usefulness is limited, and many people quickly take it too far.
Read H. A. Ironside's short book refuting the heresy of hyper-dispensationalism: