In lytic state the nucleic acid of the phage ( DNA or RNA) is not integrated in the genome of the host bacterium cell. Phage genome replicates independently of the host bacterium genome. Mature phage particles are produced and the host cell undergoes lysis. In lysogenic state the DNA of the phage is incorporated in the genome of the host cell and they are replicated together as a whole structure.

Check this resources for a better clarification:

https://courses.lumenlearning.com/microbiology/chapter/the-viral-life-cycle/

http://www.eurekaselect.com/151575/article

https://www.nature.com/articles/srep34338

Clokie MR, Millard AD, Letarov AV, Heaphy S. Phages in nature. Bacteriophage. 2011;1(1):31-45. doi:10.4161/bact.1.1.14942.

It is easier to speak with a figure. Please see the attached figure. There are two circles. The large outer circle is the 'Lytic cycle' of phage, and the small inner circle is the 'Lysogenic cycle' of the phage.

• For the lytic cycle: Phage infects bacterial cells by attaching to them, and then inject phage's genome into bacterial cells. Phage genes inside bacterial cells will produce phage proteins and assemble phage particles. Those newly formed phage particles bust out the host cells and infect other cells.
• For the lysogenic cycle: Phage infects bacterial cells by attaching to them, and then inject phage's genome into bacterial cells. However, the inject phage genome will integrate into the bacterial genome, and stay in bacterial cells peacefully, without bursting out of the bacteria cells.

If you want go deeper in the matter I recommend you this Book: "A genetic Switch", Phage Lambda Revisited, Mark Ptashne.

https://www.amazon.com/Genetic-Switch-Third-Lambda-Revisited-ebook/dp/B001WAKRQE