If a cell did not have lysosomes, it would potentially starve, since lysososmes contain enzymes important for digesting nutrients. This would be exacerbated by cells' inability to digest their own organelles, which they do sometimes when starving. Finally, in multicellular organisms, the body would have trouble destroying old, worn-out cells.
Lysosomes are tiny, membrane-bound packets of enzymes. Enzymes are special chemicals that help break down organic molecules. These break down not only food particles, but components of the cells that make them. They are so powerful that scientists still don't understand why the enzymes do not destroy the lysosomes that contain them.