Module[{x, y, ...}, expr] specifies that occurrences of the symbols x, y, ... in expr should be treated as local. Module[{x=x0, ...}, expr] defines initial values for x, ....
In[1]:=f[x0_]:= Module[{x=x0}, While[x>0, x=Log[x]]; x ] In[2]:=f[2.] Out[2]=-0.366513
HoldAll | Protected
With | Block | DynamicModule | Unique | GeneratedParameters
74th most common (1 in 1330 symbols)
44th most common (1 in 451 symbols)
100th most common (1 in 1180 symbols)
introduced in Version 2 (January 1991)