They Lytic cycle is the phage reproductive cycle that ultimately ends with the death of the host cell. A phage will attach to its host and inject it's DNA into the cell, which is then taken over by the viral DNA. The viral DNA forces the cell to produce copies of the DNA and form new viruses, which then lyse out of the cell.
The Lysogenic cycle replicates the phage genome and does not kill the cell. When the virus attaches and injects the DNA, sometimes the DNA will integrate with the host's DNA and form a prophage. The integrated DNA will replicate with the host and continuously make copies of itself. Sometimes, the virus may lie dormant forever, while other times it will switch over to the Lytic cycle and burst.