Learn Italian: Il sarto avrebbe bisogno del velluto per la gonna. - The tailor would need velvet for the skirt.

In Italian, the most common way to express "to need" is the phrase avere bisogno di (literally, "to have need of"). To say "would need", you just change the verb avere into the present conditional tense. Since the subject is il sarto (he), we use the "he/she" conditional form: avrebbe.

How do you form the conditional of "avere"? Where does the "avr-" come from?

Avere (to have) is an irregular verb in the conditional. Instead of just dropping the final -e from the infinitive, it also drops a middle vowel, changing the stem to avr-. Then, you add the standard conditional ending -ebbe for the "he/she/it" form. (Tip: This is the exact same irregular stem used in the future tense, like avrà for "he will have"!)

Is "bisogno" a conjugated verb here?

No, bisogno is a masculine noun meaning "need" (from the phrase avere bisogno di). Because it is a noun, its ending never changes, no matter who is doing the needing. Only the verb avere changes to match the subject.

Why is it "del velluto" instead of just "di velluto"?