The actual k
m of the enzyme does not change in the presence of a competitive inhibitor. However, the apparent k
m, defined by the concentration of substrate at which V = (1/2)V
max, does increase. See the following figure:
http://www.steve.gb.com/images/science/competitive_inhibition_kinetics.pngThe green trace represents uninhibited enzyme and the red trace represents enzyme in the presence of competitive inhibitor (the graph isn't entirely accurate, but it gives a good enough picture).
Irreversible inhibitors (e.g. suicide inhibitors which irreversibly bind the enzyme) will lower the effective [E]. If all of the enzyme is bound by a suicide inhibitor, then your V
max will go to zero. A suicide inhibitor will not change the apparent k
m of the reaction.
For other types of inhibitors, you can see different effects depending on the mechanism of action of the inhibitor (e.g. does the inhibitor bind the enzyme alone, does it bind the enzyme-substrate complex, does it have equal affinity for apo-enzyme and enzyme-substrate complex?)
(note: I'm moving this to the biochem forum since it's more about biochem than organic chem)