Competitive inhibitor competes for the active site. Therefore it will interfere with the binding of the substrate. Therefore it increases the apparent K_M.

A strictly non-competitive inhibitor does not compete for the active site. It however inhibits the catalysis by reducing the available molecules of active enzyme, E₀ (if it is a perfect inhibitor), thereby lowering the V_max.

There can be mixed inhibitors too and there can be different kinds of mixed inhibition. For example an inhibitor that interferes with catalysis and can also compete for the active site.

Depending on how you define K_M, a non-competitive inhibitor may or may not change it. If such inhibitor can bind to the enzyme at non-active site and reduce its k_cat, it changes the K_M as well as V_max in case of Briggs-Haldane kinetics.

Competitive inhibitor competes for the active site. Therefore it will interfere with the binding of the substrate thereby increasing the apparent K_M.

A strictly non-competitive inhibitor does not compete for the active site. It however inhibits the catalysis by reducing the available molecules of active enzyme, E₀ (if it is a perfect inhibitor), thereby lowering the V_max.

There can be mixed inhibitors too and there can be different kinds of mixed inhibition. For example an inhibitor that interferes with catalysis and can also compete for the active site.

Depending on how you define K_M, a non-competitive inhibitor may or may not change it. If such inhibitor can bind to the enzyme at non-active site and reduce its k_cat, it changes the K_M as well as V_max in case of Briggs-Haldane kinetics.