Identifying type of inhibitor from $K_m$ and $V_{max}$
Competitive inhibitor competes for the active site. Therefore it will interfere with the binding of the substrate thereby increasing the apparent KM. 

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 Vmax.

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 KM, 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 kcat, it changes the KM as well as Vmax in case of Briggs-Haldane kinetics.