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:

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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

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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

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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

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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

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

In the latest segment of Le Jour où tout a basculé - J'ai piégé mon fan, Alex uses a phrase whose meaning may surprise you:

Mais bon, c'était pour la bonne cause. Tu m'étonnes. Regarde.

But OK, it was for a good cause. You're not kidding. Look.

Captions 7-8, Le Jour où tout a basculé - J'ai piégé mon fan - Part 7

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The literal translation of tu m'étonnes is "you surprise me," but it's often used as a set phrase meaning "you're not kidding," "no kidding," or "tell me something I don't know." Used in this way, it has the opposite meaning of its literal translation—the person is not surprised by what they just heard. Tu m'étonnes is very similar to the English expression "surprise, surprise," which is also used ironically to convey a lack of surprise. 

Sans blague is another phrase meaning "no kidding" or, more literally, "no joke." This one, however, can express surprise:

Je suis né le 3 novembre. -Sans blague! Moi aussi!
I was born on November 3. -No kidding! So was I!

The verb étonner has the same root as the English verb "to stun." It means "to surprise," "astonish," or "amaze":

Sur l'eau, il vit son reflet, totalement étonné

In the water, he saw his reflection, totally surprised

Caption 29, Contes de fées - Le vilain petit canard - Part 2

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Les héritiers de Jules Verne n'ont pas fini de nous étonner.

Jules Verne's heirs have never ceased to amaze us.

Caption 26, Le Journal - Le record du Tour de Monde!

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And the English "surprise" comes directly from the French surpris(e):

Je suis un peu surpris.

I'm a little surprised.

Caption 38, Lea & Lionel L - Le parc de Bercy - Part 1

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Unsurprisingly, the verb surprendre means "to surprise":
 

Tu vas mener l'attaque pour les surprendre.

You're going to lead the attack to surprise them.

Caption 28, Il était une fois - les Explorateurs - 15. Bruce et les sources du Nil - Part 2

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But it can also have the related meaning "to catch," "come upon," or "discover":

Louise surprend René et Edna en pleine conversation.

Louise catches René and Edna deep in conversation.

Caption 2, Le Jour où tout a basculé - Mes grands-parents sont infidèles - Part 8

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Just as there are two words for "to surprise" (étonner and surprendre) and two words for "surprised" (étonné[e] and surpris[e]), there are two words for "surprising": 

C'est pas étonnant que beaucoup de peintres soient venus s'installer ici sur Arles.

It's not surprising that many painters came to settle here in Arles.

Caption 12, Arles - Un Petit Tour d'Arles - Part 3

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C'est un endroit vraiment surprenant en plein cœur de Paris.

It's a really surprising place right in the heart of Paris.

Caption 14, Voyage dans Paris - Les Secrets de Belleville

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Can you guess what la surprise and l'étonnement mean? Surprise, surprise!