páirtí is a transliteration of the English word "party" in the sense of "a political party - páirtí polaitaíochta, or person/group that is "a party" in a legal sense: "an interested party" - páirtí leasmhar.
cóisir is the Irish word for a celebration, what is called "a party" in English, even though the word "party" means something else.
No, the hints do not say "Is the party tonight".
The hints for the individual words may be "is", "the", "party" and "tonight", but the word order in Irish isn't the same as it is in English, so you can't put the hints together in the same order. The basic structure of an Irish sentences is verb-subject-object, whereas English uses subject-verb-object.
I realize that this is an awkward sentence, but there are contexts in which it would make sense to say "There's the party tonight."
Such as reminding someone that a specific event is happening. "What were we doing tonight again?" "There's the party tonight. Did you forget?"
I translated the above exercise as "There is the party tonight," & it was marked wrong. If there's no meaningful distinction between the two, then both should be accepted.
If there is a distinction, then how would one say "There is the party tonight"?
To use "there's" as an indication of existence, you would use the existential ann, as in Tá Dia ann - "there is a God", Tá an saol ann - "the world exists". But because ann is ambiguous for words like cóisir that occur in a specific location, the ann in tá an chóisir ann would usually be understood as the locational ann ("the party is there"), so the short answer is that you wouldn't say "There is the party tonight" in Irish. You would simply address the matter in a different way - Céard faoin gcóisir sin?.
For an Irish speaker, tá an chóisir anocht only means "the party is tonight" (in the sense of "tonight as we speak", not "tonight after I get off work").