What cell parts are common to all of these cells

If you're talking about Prokaryotic and Eukaryotic cells...

They both have a cell membrane, which is basically like the wall of the cell.

They both have ribosomes, which are basically protein things.

They both have nucleoids, basically DNA.

They both have cytoplasm, or the stuff that fills in the remaining space in the cell.

If you are talking about animal and plant cells...

Animal cells basically share everything they have with plant cells, apart from the fact that plant cells have cell walls instead of a cell membrane. 

However, plant cells don't have the flagella, centrioles, and a few more other things that animal cells do.

Plant cells also have a central vacuole and chloroplasts.