"The summer is a season."
Translation:Is séasúr é an samhradh.
So, I'm going to describe this in the way my class was taught. It'll be watered down, because y'know, we were first semester Irish students without much grammar knowledge. So, your word order is going to be VANTP.
V = verb
A = Indefinite nouns
N = Names
T = Definite nouns
P = Pronouns
So that's your general order. If you use a definite noun, a proper noun, or a name, you need to include the pronoun (é, í, iad, srl - note, after ní, these prefix an "h") before the first one.
So in your first sentence you have and indefinite noun (araicnid) and a definite noun (an damhán alla). So, putting these in order (A before T), you get Is araicnid an damhán alla. However, you need the pronoun before the definite noun. Since damhán alla is feminine, you use í. This giving you Is araicnid í an damhán alla.
Your second one you have a name (Pól) and a definite noun (uachtarán na hÉireann). So, putting those together, you get Is Pól uachtarán na hÉireann. Yet, you still need that pronoun. Is é Pól uachtarán na hÉireann.
Hope that helps explain it a little. And note, this is really basic, and barely scratches the surface of the copula.