It really depends what the mixture consists of. You would not use the same method to separate sand from gravel as you would to separate salt water from pure water.
Some common separation methods include:
Filtration: works best for solids suspended in a liquid (to separate sandy water into dry sand and pure water for example) or for two solids with very different particle sizes (to separate a mixture of sand and gravel).
Distillation: works to separate liquids that have different boiling points. This method is used to separate alcohol from water (to make hard liquor) and the essentially the same method is used to take oil from the ground and turn it into petroleum products such as gasoline.
Chromatography: used to separate mixtures of liquids based on polarity. Many different types exist, including thin layer chromatography (TLC), gas chromatography (GC), and column chromatography.
Extraction: used to separate a mixture of solids based on their solubilities in organic and inorganic phases. Differences in pH can also be used to separate different solids by selectively ionizing one compound, causing it to dissolve more readily in inorganic solution.
See the Related Questions to the left for more information.