What type of protein controls what goes in and out of cells?

Integral Membrane Proteins control the movement of substances into and out of the Cell (that cannot pass directly through the bi-lipid layer).

These are proteins that span the width of the membrane [40 nanometers], and can be channels, pumps, gates of many descriptions, co-transporters, etc.

As proteins go, these trans-membrane proteins have internal and external hydrophyllic lengths that have immensely important and varied bio-functions, and the trans-membrane segment is of course hydrophobic.