Learn Russian: Ему нужно подписать этот отчёт сегодня.

In Russian, obligation is usually expressed impersonally. Instead of saying 'he needs', the structure literally translates to 'to him it is necessary'. This is why the subject Он (he) changes to the Dative case pronoun Ему (to him).

Could I use надо instead of нужно in this sentence?

Yes, you could absolutely say Ему надо подписать. When followed by an infinitive verb like подписать, both надо and нужно mean 'needs to' or 'it is necessary to' and are completely interchangeable in this context.

Doesn't the verb подписать need a direct object? Why does этот отчёт look like it's in the Nominative (dictionary) case?

It actually is in the Accusative case, which marks the direct object receiving the action! However, because отчёт (report) is a masculine inanimate noun, its Accusative form is identical to its Nominative form. The modifying word этот (this) follows the exact same rule.

Why use the perfective verb подписать rather than its imperfective pair?

The perfective form подписать is used because we are talking about a single, one-time action that needs a successful result today. If it were a repeated, routine action like 'he needs to sign reports every Friday', you would use the imperfective form.