Have you been given definitions of what it means for a function to have certain singularities? If so, what is it about the definitions you don't understand?

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