Centrifugation
Generally the first step after forming a crude extract is a simple filtration or centrifugation to remove the large material. Centrifugation is a process that involves the use of the centrifugal force for the sedimentation of mixtures with a centrifuge. This process is used to separate two immiscible liquids with more-dense components of the mixture migrate away from the axis of the centrifuge, while less-dense components of the mixture migrate towards the axis. Centrifugation alters the effective gravitational force on to tube/bottle so as to more rapidly and completely cause the precipitate ("pellet") to gather on the bottom of the tube. The remaining solution is properly called the "supernatant". The supernatant liquid is quickly decanted from the tube/bottle without disturbing the precipitate.
Differential centrifugation, as shown in the figure, is multiple rounds of centrifugation at increased speeds and time allows for different cellular fractions to be separated.
At this point, however, an assay is key because is your protein of interest?!