The grammar
Much as 古手梨花 described, ので is used to express a reason for something, similar to "because" in English. Grammatically, ので is the conjunctive conjugation of のだ (plain form) and のです (polite form), which can be parsed as possessive or genitive の + the copula ("to be") verb だ or です.
の attaches directly to nouns or noun phrases, but because のだ describes reasons, it attaches to statements. If you're making a statement about a noun, you finish with the copula だ or です. And, grammatically, those statements that のだ attaches to must be in the 連体形【れんたいけい】 or "attributive" form, and the attributive form of だ or です is な.
Examples:
- 本です。 (It's a book.)
⇒ 本な[の]{●}[で]{●}[す]{●}。 (It's because it's a book.)
⇒ 本な[の]{●}[で]{●}読みます。 (I read it because it's a book.)
(Note the reversed clause order to produce a more natural English translation.)
Your sample sentence
今日は日曜日なので銀行は休みです。
Basically, this is a longer sentence with a shorter sentence embedded. Let's break this down like the simpler example further above.
- 今日【きょう】は日曜日【にちようび】です。 (Today is Sunday.)
⇒ 今日【きょう】は日曜日【にちようび】な[の]{●}[で]{●}[す]{●}。 (It's because today is Sunday.)
⇒ 今日【きょう】は日曜日【にちようび】な[の]{●}[で]{●}銀行【ぎんこう】は休【やす】みです。 (The bank is closed because today is Sunday.)