Learn Portuguese (Portugal): Preciso de um par de sapatos novos.

The verb precisar (to need) is almost always followed by the preposition de when you mention what you need. In English, we just say "I need a pair", but in Portuguese, it translates more like "I have need of a pair", which is why we say Preciso de um par.

In English it says "a new pair", but in Portuguese "novos" is plural. Why doesn't it agree with the singular word "par"?

In English, we usually describe the pair itself as new ("a new pair of shoes"). In Portuguese, it is much more natural to attach the adjective directly to the shoes. So novos is masculine and plural to agree with sapatos, literally meaning "a pair of new shoes."

Why does "novos" come after "sapatos" instead of before it?

In Portuguese, descriptive adjectives like novos (new) almost always come after the noun they are describing. This is the opposite of English, where the adjective comes first.

Could I just drop "um par de" and just say "I need new shoes"?

Yes, you can! But remember that you still need the preposition de after the verb. You would say Preciso de sapatos novos. Using um par de just emphasizes the matched set, exactly like "a pair of" does in English.