C'e & Ce n'e
They both mean (there's) but what's the difference ?
I tell you some individual details:
1) Ci (c'è = ci è) works as an adverbial particle (there) indicating the existence of something or the presence of someone or something in a certain place, e.g. Ci sono tanti tipi di animali nel nostro ecosistema. (There are many types of animals in our ecosystem.), C'è un cane in cortile. (There is a dog in the yard), it is like a kind of subject of the verb. As a consequence in Italian it's not possible to omit ci (there), for example in questions, as in English, e.g. Chi c'è sulla barca? (Who is [there] on the boat?). Ever. In that way you would eliminate the concept of presence in that place.
Note: further ci can indicate a place in other constructions (here/ there) e.g. Quando posso trovarti a casa? Potrei esserCI domani. = Potrei essere qui/lì domani. (When can I find you at home? I might be here/there [= at home] tomorrow.), Io CI sono. = Io sono qui. (I'm here); Io CI vado. = Io vado lì/là. (I go here/there.); sometimes this particle even changes the meaning of some verbs e.g. Io CI ho pensato = Io ho pensato a qualcuno (I thought about someone - here the concept is "to turn your mind towards a figurative place, in this example a person), different from Io LO penso = Io penso lui (I think about him, like "to imagine/represent him in my mind", less emotional than the first one).
You can find other functions of ci here, there are many comments.
2) Ce (ce n'è = ce ne è) is like a "brother" of ci because ci is turned into ce every time it is followed by a clitic pronoun (pronomi atoni) - lo, la, le, li - also including the pronominal particle ne (which has various functions, it introduce many complements); ce n'è indicates the presence of something more specific ,because someone mentioned it previously: C'è la tua torta? Certo! Ce n'è abbastanza per tutti. (Is there your cake? Of course! There's enough of that for everyone.)
3) let's remember the plural forms: ci sono and ce ne sono.
Be careful, I' ve seen someone to write c'è ne or c'è n'è, these versions are wrong. :)
C'è = There is. It needs a subject: 'C'è un libro sul tavolo'
Ce n'è: We know what we are talking about. Es: Domanda: 'C'è burro?' --> Risposta: 'Sì, ce n'è.
N'è comes from 'ne è' --> plural: (Ci sono libri in casa?) Risposta: 'Sì, ce ne sono'
C'è (Ci + è) = There is.
Ce n'e (Ci + ne + è → Ce ne è → Ce n'è) = There is of it.
Ce, the ancient form of the pronoun ci (from the Latin ecce hīc) that is preserved in union with the pronouns atoni lo, la, li, le and particle (see sentences as there are no more, I placed it last night) , from there (there is); n'è (or is it) from the simple adverb or personal pronoun ("with reference to people already named, as a complement of argument, specification and partitive, with the meaning of him, her, them: is very linked to father, he always speaks of it with affection: parents were missing, there was none at the school "), or no negative coordinating conjunction, which is often, inter alia, often written with the grave accent, * nor, instead of the acute accent.