Viruses that enter the lytic cycle do so by taking over the regulatory machinery of the cell, expressing genes that are encoded by the virus, which themselves are more copies of the virus. The virus then becomes to prolific in number within the cell that the cell wall bursts (Lyses).
The opposite are viruses that encode themselves into the hosts DNA, which is referred to as the lysogenic cycle. A virus can basically sit in your genome for any period of time and then at any point, usually triggered by stress, enter the lytic cycle by taking over the regulatory mechanisms in the cell and forcing it to produce large quantities of the genetic material that has been embedded into the hosts DNA, which are almost always the components of more of the same virus.
Again and again the answers come down to terminology. Lytic means to break. The opposite is lysogenic which means prevented from moving; held down or secured.