Both cycles are methods of viral reproduction.  Both cycles involve the introduction of the virus into a cell to use the cell's genetic material to replicate more viruses.  The key difference is that in the lytic cycle the viral DNA is maintained in a separate pocket from the cellular DNA and matures separately while in the lysogenic cycle the viral DNA is kept mixed with the celluar DNA.  Another key difference is that in the lytic cycle the viruses replicate constantly which results in the eventual lysis, or death, of the host cell to release the new viruses while in the lysogenic cycle the virus replicates only with the regular cellar mechanism for reproduction and is only released through some separate outside event.