Normally inhibition possibly can be differentiate into various category but when you go through different books or articles sometimes we get confused in these terminologies
You are talking about two different enzyme kinetics here. When referring to enzymes obeying Michaelis–Menten kinetics then competitive, non-competitive and uncompetitive can be differentiate by the Lineweaver–Burk plot. Competitive inhibition affects substrate binding site so Km is affected(changes). Non-competitive inhibition does not affect the substrate binding site so Km is constant but Vmax is affected(changes). However in uncompetitve inhibition both the substrate binding site and the other side are affected so both Km and Vmax are affected (changes).
For allosteric enzymes, since they are made up of more than one subunits and have allosteric binding sites that can influence the substrate binding sites, they do not obey Michaelis–Menten kinetics and cannot be differentiated by the Lineweaver–Burk plot.
As Masood said, if one varies the substrate and inhibitor concentration, and then does reciprocal plots of initial velocity vs. substrate concentration as a function of inhibitor concentration, the plot will tell you what type of inhibition you have. Non competitive can be allosteric or not. It is when I or S binds to E to form ES or EI, and they compete for binding, and I binds to ES to form ESI, but I can bind at the same site or another site as S. The most common kind is when I binds another site. I think for uncompetitive, I just binds to ES to form ESI, and the best fit to the plots determines the type of inhibition. I always forget what plots determine what type of inhibition, but Masood has it above.
You are talking about two different enzyme kinetics here. When referring to enzymes obeying Michaelis–Menten kinetics then competitive, non-competitive and uncompetitive can be differentiate by the Lineweaver–Burk plot. Competitive inhibition affects substrate binding site so Km is affected(changes). Non-competitive inhibition does not affect the substrate binding site so Km is constant but Vmax is affected(changes). However in uncompetitve inhibition both the substrate binding site and the other side are affected so both Km and Vmax are affected (changes).
For allosteric enzymes, since they are made up of more than one subunits and have allosteric binding sites that can influence the substrate binding sites, they do not obey Michaelis–Menten kinetics and cannot be differentiated by the Lineweaver–Burk plot.
Allosteric basically just means "other" site and is a general term for when something binds a site other than the active site. An example of an allosteric enzyme is hexokinase, that is regulated by binding of AMP, but I believe it still follows Michelis-menten kinetics. But if AMP is the inhibitor, and it binds with ES to form ESI, then AMP would be a non competitive inhibitor of hexokinase as well as allosterically regulating the enzyme.
@Marcia Moss With all respect, if by "hexokinase" you mean "glucokinase", it actually does not follow MM kinetics (see the article below).
@Masood Sepehrimanesh With all respect, your 3rd claim ("Mix inhibitor: Both Km and Vmax were decreased.") is not correct, and you can easily see it in your graph (see my drawing below). Your claim describes uncompetitive and not mixed inhibition. The correct claim for mixed inhibition would be: vmaxapp is decreased and Kmapp is increased.