Not sure what you are looking for with this. I'll give you an explanation of both:
This is about viruses. We can use bacteriophage (a bacterial virus) to explain the difference:
Lytic infection: the phage infects a bacteria, once it's genetic material is inside, it gets replicated, proteins are expressed from the phage DNA that code for it's structure and the new phage genomes get packaged into new structures...when there are enough of them, more proteins are expressed that lyse (pop) the cell open, thereby spewing out the new phage who can go off and infect more bacteria. Some phages are strictly lytic and this is all they can do.
Others are what we call temperate. They can either cause a lytic infection or they can go lysogenic (we call these temperate phages).
Lysogenic:
Lysogeny is when a temperate phage (e.g. lambda) infects a bacteria but instead of immediately making more copies of its genome, integrates its genome into the chromosome of the bacteria. For a phage like lambda, it integrates at a very specific spot in the bacterial chromosome (called attB). In this manner, the phage is replicated along with the bacteria. At some point the phage can decide to go lytic - it undoes itself from the bacterial chromosome and then like a lytic phage replicates its genome, packages it and then lyses the cell.
Good examples of human lysogenic infections are Varicella that causes chicken pox or herpes simplex I - the virus that causes cold sores.