Note that nessun is used as an adjective, and must agree in gender and number with the thing that it describes. Moreover, nessun and niente usually follow the verb when they act as the object. When one of these words is the subject of the sentence, its position can vary:

Nessuno parla. (No one speaks.)

Neinte ci piace. (We like nothing at all.)

Non parla nessuno. (No one speaks.)

To review the negatives, compare the two sets of phrases in the following table:

POSITIVE AND NEGATIVE PHRASES
 
POSITIVE
 NEGATIVE
 
Ha fame. (She is hungry.)
 Non ha fame. (She isn’t hungry.)
 
Ho comprato qualcosa. (I bought something.)
 Non ho comprato niente. (I didn’t buy anything.)
 
Ride sempre. (He always laughs.)
 Non ride mai. (He never laughs.)
 
Vediamo qualcuno. (We see someone.)
 Non vediamo nessuno. (We don’t see anyone.)