Nem and Nincs become Sem and Sincs when meaning "Neither ...". For example:
A - "Nem akarok iskolába menni" (I don't want to go to school)
B - "Én sem" (Neither do I)
A - "Sajnos nincs pénzem" (Unfortunately I don't have any money)
B - "Nekem sincs" (Neither do I)
Note that you can also say "Se" instead of "Sem" ("Én se", for example)
Nem simply means "not". Nincs, on the other hand, is a verb. It's the lovechild of nem and van, so whenever you would write "nem van", you use nincs instead. It means "there is no", or if related to locations or times, "is not". It's the negation of van, of existence. It also has a plural form, nincsenek - "there are no" or "are not".
Sem and sincs have the same relation, but sem is a combination of "is nem", meaning "not either". As a result, sincs means "there is no... either" or "is not... either".