Dispensationalism basically suggests that at different ages (eras or dispensations) in human history, God has interacted with people in different ways, through various covenants:
Antedeluvian(consciousness - people did what they wanted)
Civil Government(After the flood, the rainbow covenant was for all people)
Patriarchic (Abraham and the particular family He called)
Mosaic Law (God makes a nation and puts his presence there)
The Church Era (The Holy Spirit dwells in His people)
Note that there are seven dispensations. A perfect week, and the last dispensation corresponds with Rest. In each of the dispensation, God institutes different systems of relating to His people. He walks in the garden with Adam, He makes a covenant with all of creation, then one man, then a nation. He then establishes a throne, then a new covenant in Jesus. Today, we interact with the Holy Spirit in a way that other men and women of God did not before Pentacost. Soon we will see Him face to face.
Controversy around dispensationalism usually happens when dispensationalist take the different covenants and institutions of God and say that for each era, the people in that era are saved through the system of that era. For example, a follower of God during the Mosaic covenant is saved through the law. However, Paul reminds us in Galations 3 that even Abraham was saved through faith. All of God's people are saved through faith in Him, always through His Son.