What are the methods used to separate component of a mixture?

There are many ways, I'll probably miss some!
Crystallization: A mixture of solids is dissolved in a hot solvent, concentrated and allowed to cool. The solids slowly become insoluble and crystallize. Usually, a major component will crystallize in rather pure form, leaving others behind in solution.

Distillation to separate by boiling point.

Extraction: You can adjust the conditions to alter solubility of various components in water and organic solvent. For example acids are soluble in water when the pH is basic, and soluble in organic solvent when it is acidic.

Reactions: you can make other compounds (temporarily) of components in the mixture to separate them. This is most often done to separate inorganic materials by precipitating and extracting their oxides, sulfides, chlorides, nitrates,etc. or by dissolving them selectively in various acids.

Chromatography: Different things in a mixture will adhere to a wide variety of certain materials such a silica gel and paper. There are many chromatographic techniques including gas and liquid types of chromatography. As the mixture in solution is passed over a the material, interaction with it will slow down the rate of passage of each component. If two components are slowed to sufficiently different degrees, one will completely come off the material before the next begins, completely separating the two. This has become the most common way (or ways, because there are so many different ones) to separate components of mixtures.