Let us start with elements. Atoms from an element have a set amount of protons, a variable amount of neutrons, and a set amount of electrons. Atoms of an element help define the physical and chemical properties of those elements. The number of protons and electrons are what arrange the different elements into groups and families on the periodic table of elements. Hydrogen, chlorine, helium, and gold are examples of elements.
Two or more elements that are chemically combined make up a compound. These elements are chemically combined with each other by gaining, losing, or sharing electrons. It is interesting to note the compounds physical and chemical properties are usually strikingly different from the elements that make them up. A good example would be the formation of water from hydrogen and oxygen. Hydrogen is very explosive when combusted with oxygen, and oxygen is what allows things to combust, or burn. Together, they make a compound that puts fires out, water.
A mixture is combination of two or more elements that are physically combined, meaning they may be separated by ordinary physical means. The mixture may not have strikingly different physical and chemical properties, as the individual components that make it up still retain their identities as substances. This is like ordering a supreme pizza, then picking off the parts you don't like.