"The boy and the girl."
Translation:An buachaill agus an cailín.
The suffix ín is a diminutive meaning little or small. When you just tag it onto an existing word it doesn't change the gender (so an bheanín for "the little woman"), but when it is an integral part of the word, it's a masculine ending. Not all words that end in ín are using it as a dimunitive, though, so braillín is a feminine word, for example.
gasúr is an alternative that is sometimes used for buachaill, though it might be more usefully compared to "lad" or "youngfella". It's close counterpart, garsún has obvious links to the French "garçon".
When you submit an incorrect answer, Duolingo does a crude alphabetic comparison with the list of acceptable alternative answers, and it decided that you answer was alphabetically closest to an acceptable alternative answer that used gasúr rather than buachaill. If you look at the default translation shown at the top of the webpage for this discussion, you will see buachaill.
That's a bit like asking "when do you use a 'b'?". á and a are two different letters, and words that are spelled with a á or é or í or ó or ú require you to use the accented letter.
The accented vowels don't sound the same as the unaccented vowels, so you can tell which one is used to spell a word.
Accents aren't like lenition or eclipses that come and go depending on grammatical rules. Accents don't change and they don't tell you anything about the grammar - the only rule for using an accented vowel is that you need the sound that it makes.