"an t-sràid ghoirid"
Translation:the short street
While I wouldn't entirely trust Wiktionary as a source, their article on "an" seems to have a good chart of all its inflections.
Assuming it is correct, then the answer to your question is that you place a "t-" after "an" when the following noun is masculine and begins with a vowel in the nominative case, is masculine and begins with s+vowel, sl, sn or sr in the dative and genitive cases, or is feminine and begins with s+vowel, sl, sn or sr in the nominative and dative cases.
sràid is a feminine noun beginning with sr and this is the nominative case, so it gets a "t-".