Strainers are devices for getting solids out of liquid. A contrasting device is something such as a colander, which instead focusses on getting the liquid out of or off of a solid.
Sieves are devices for separating differently sized solid particles from one another: e.g. separating the coarser bran particles from wheat flour. It can loosen and aerate, such as when sifting flour before using it in baking. The act of using a sieve is called either “to sieve” or “to sift”.