Animal cells - All animals are made up of cells, these cells are 'eukaryotic', meaning that they have a nucleus and other structures that are surrounded by membranes.
Nucleus - The nucleus contains genetic material, including DNA, which controls the cell's activities.
Cytoplasm - A jelly-like substance that contains dissolved nutrients and salts and structures called organelles. This is where a lot of reactions take place. This can be seen under a light microscope.
Mitochondria - This is where cellular respiration occurs, it takes in nutrients from the cell, breaks it down and releases it as energy.
Cell Membrane - This is a layer around the cell that is permeable to some substances, therefore controlling what enters and leaves the cell.
Ribosomes - These are very small structures where protein synthesis occurs.
Plant cells - Plant cells are also eukaryotic, meaning that they have a nucleus and other structures that are surrounded by membranes.
Animal and plant cells have several structures in common (listed above), there a few that are specific to plant cells and animal cells, however:
Cell wall - Provides structure and protection for the cell, only plant cell walls are made from cellulose.
Chloroplast - This is the green organelle that absorbs light energy for photosynthesis, it also contains the enzymes required for photosynthesis.
Permanent vacuole - This is filled with cell sap to keep the cell swollen (turgid)
Image credit - Wikibooks