I seem to be seeing both Nèill and Niall in the Gaelic answers in various places. I put Nèill here, and was marked wrong, but I have seen that marked right elsewhere for what was apparently the same name. Is it two different names? Is it the same name in two different cases? Nominative and vocative, or vice versa? I'd appreciate an explanation.
Yes it's exactly that.
Nèill is the vocative casing of Niall. It looks a bit of an unusual mutation compared to Seumas (for example), which mutates to Sheumais. Vocative causes lenition and a slender ending. Words starting with N never lenite in writing as Nh would be impossible to pronounce in Gaelic. I imagine at some point in history the spelling became Niaill and then mutated to Nèill.
It can be a bit confusing which case is which. Whenever a case change is marked by slenderization (for the masculine vocative, or any other reason you will learn about later), the following rules apply
+It only happens to words that end in a broad consonant
+You get an i before the final consonant. This is the characteristic feature.
+What happens to the rest of the vowel/diphthong is a bit of a guess
cat → cait
fear → fir
Niall → Nèill
bòrd → bùird
mac → mic
So recognising a slenderized form is easy, but knowing which is correct is difficult.
To cheer you up, Mark lists 22 possibilities for nouns.