In the lytic cycle, a virus that has invaded and entered a cell takes over the cell's replication mechanism. The virus makes viral DNA and viral proteins.
The virus then lyses (breaks open) the cell membrane, allowing the newly produced viruses to leave the now disintergrated host cell to infect other new cells.
In a lysogenic cycle, the viral DNA or RNA enters the cell and integrates into the host DNA as a new set of genes called prophage. The prophage is passive and non virulent and does not destroy the cell. However if the prophage undergoes any stress or mutation or is exposed to UV radiation, it may adopt the lytic cycle and become infective.