Look at this easy way to remember:

Masculine singular nouns:

These are like the indefinite articles in English:
A cat
An apple

Look at Irish:
An cat
An t-éan

If you remember that t-prefixing is like keeping the n in the indefinite article in English before an initial vowel, it becomes easy to remember


Feminine singular nouns:

These are easy to remember because (a):

Initial lenition of consonants:
An ghloine

(b): t-prefixing is a form of lenition, so it is natural to have it on the s in the feminine noun set

An tsúil


Masculine singular genitive:

An chait
An éin
An tsagairt

Here it is natural to have the t prefix to the s as t-prefixing is lenition in another form. No prefixing to the vowel as no lenition can happen there.


Feminine singular genitive:

The feminine article na is used and causes no initial change except for before vowels as this is to stop an hiatus event


Masculine plural:

The plural article na is used and again causes no initial change except for before vowels to stop an hiatus event


Feminine plural:

It is the same above as gender diffracted by case does not matter in the plural in Irish


Masculine genitive plural:

The old genitive plural article nan/nam survives in the form of na plus the initial eclipsis of the following consonant or n-prefixing to vowel:

Na bhfear
Na n-éan
Na sagart


Feminine genitive plural:

The same as above: the old genitive plural article nan/nam survives in the form of na plus the initial eclipsis of the following consonant or n-prefixing to vowel:

Na gloiní
Na súl
Na n-áiteacha