The most important organelle that distinguishes plant cells from animal cells is the chloroplast, which contains chlorophyll. Chlorophyll captures sunlight in the chloroplast, and the energy from sunlight (along with water and carbon dioxide) is subsequently converted into glucose. This process is known as photosynthesis, and only plants (along with some protists and bacteria) are able to do this. Chloroplast contains its own unique DNA; DNA in animal cells are found in the mitochondria (plants also have mitochondria with DNA that programs non-chloroplast organelles).
Plant cells also have an organelle called a cell wall which plays an important role in photosynthesis. The cell wall is on the exterior of the cell, forming an additional layer beyond the cellular membrane. The wall is rigid, but is also permeable to water. During photosynthesis, water enters into through the cell wall, but the wall remains rigid enough so that the cell does not expand due to the presence of water. Some plants (such as trees) also have a lignan, which is a secondary cell wall for extra toughness.
Another organelle that is exclusive to plant cell is the large central vacuole, which is the largest organelle, by volume, in the plant cell. The primary purpose of the large central vacuole is to store excess water and waste products. The large central vacuole displaces the other organelles to the periphery of the plant cell, which in turn keeps them protected from potentially harmful substances. Animal cells do have vacuoles, but not the same size as a plant's.
One organelle that is unique to animal cells (and certain types of protists) is the centriole, which allows for the cell to move. During mitosis (the process in which a cell divides and creates new cells), centrioles line up at the edge of the cell membrane. In protists, centrioles develop into flagellate (a whip-like apparatus) or a psuedopod (Greek for "false foot") to allow movement; some animal cells still use the centriole for movement, such as the tail found on sperm cells.