De nouveau and à nouveau both mean "again" (or more literally, "anew"), and you'll often find them used interchangeably in everyday speech. But technically there's a subtle difference between them. De nouveau implies a repetition of something that already happened:

BANNER PLACEHOLDER

Le lendemain il se retrouva de nouveau sur le bord d'un immense lac.

The next day, he found himself again on the edge of an immense lake.

Caption 13, Contes de fées - Le vilain petit canard

 Play Caption

Je ne vous ai pas entendu. Pourriez-vous m'expliquer de nouveau?
I didn't hear you. Could you explain it to me again [repeat what you just said]?

On the other hand, à nouveau implies something happening in a different way than before—that is, in a new way: 

On retravaille à nouveau l'orthographe français [sic: française].

French spelling has once again been reworked.

Caption 46, Le saviez-vous? - L'histoire de la dictée

 Play Caption

Je ne comprends pas. Pourriez-vous m'expliquer à nouveau?
I don't understand. Could you explain it to me again [in a different way]?

Do you see the difference between the second sentences in the examples above? If you don't hear something someone said, you want them to repeat it. So you use de nouveau. But if you don't understand what they said, you want them to rephrase it, say it in a new way. So you use à nouveau.

Note that both these expressions only use nouveau, not the other forms of the adjective (nouvel, nouveaux, nouvelle, nouvelles). If you see any of these after de, you're dealing with "new," not "again":  

...et de la mémorisation de nouveaux mots ou de nouvelles phrases.

...and the memorization of new words or new phrases.

Caption 49, Le saviez-vous? - Les bénéfices de la dictée

 Play Caption

If you forget when to use à nouveau versus de nouveau, you can always just use encore, the most basic equivalent of "again":
 

On espère te... te voir encore sur d'autres scènes en Alsace?

We hope to... to see you again on other stages in Alsace?

Caption 62, Alsace 20 - Femmes d'exception: Christine Ott

 Play Caption

BANNER PLACEHOLDER

Just keep in mind that encore can also mean "still," as we discussed in a previous lesson.