The first few steps in both the lysogenic and the lytic cycles are similar: the virus enters the host cell and integrates its own genetic material into that of the host's. what happens after that is different. In the lysogenic cycle, the virus does not destroy the host cell, but replication of the infected host cell results in daughter cells that contain the viral DNA. On the other hand, viruses that undergo the lytic cycle work by replicating inside the host cell by taking over the cell processes and mechanisms, and then the host cell is destroyed and the new viruses can then infect other cells. For an illustration of the difference between the two viral cycles, you can visit this site: http://biology.kenyon.edu/HHMI/Biol113/2virus.htm
During the lytic cycle, the virus kills the host cell using the host cell's own defenses after it has reproduced itself. In the lysogenic cycle, the virus reproduces without killing the cell. Look here for more information: http://staff.tuhsd.k12.az.us/gfoster/standard/bvirus.htm.