"I have a rat and a mouse."
Translation:Tha radan agus luch agam.
If you compare the English sentence "x has y" with the Gaelic sentence tha y aig x, you can see that the subject of the sentence in English (x) is the object of the sentence in Gaelic, and vice versa.
In tha radan agus luch agam, agam = [aig + mi], so mi is already there in the sentence (although it's in disguise). Literally, "a rat and a mouse are at me". As the sentence already has a subject (a rat and a mouse), adding another one (I) doesn't make sense. If you were to say tha mi radan agus luch agam you would end up with something like "I a rat and a mouse are at me".
agam is an example of a prepositional pronoun - these combinations of a preposition and a pronoun are very important in Gaelic.
The list of prepositional pronouns with aig is:
agam [aig + mi] = at me
agad [aig + thu] = at you (singular, informal)
aige [aig + e] = at him/it
aice [aig + i] = at her/it
againn [aig + sinn] = at us
agaibh [aig + sibh] = at you [plural or singular formal]
aca [aig + iad] = at them
So if you wanted to say "you have a rat and a mouse" instead of "I have a rat and a mouse", you just substitute agad for agam - tha radan agus luch agad, and so on with the other prepositional pronouns.
If you are using a name rather than a pronoun, e.g. Iain has a rat and a mouse, you have tha radan agus luch aig Iain.