I can give you definitions of the two types of singularities you asked about, from these it should be obvious what the difference between them is.
In terms of a Laurent series;
has a removable singularity at a point a if in it's Laurent series,
we have that
has an essential singularity at a if
for infinitely many
Now, about 0,
Substitute z for 1/z and then look at coefficients of negative powers of z, you should then see what type of singularity occurs at z=0