While preparing a gâteau aux pommes with Marie, Jeremy uses the phrase grâce à several times when noting the utensils they use to add the ingredients:

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On ajoute cent grammes de sucre mesurés avec précision grâce à un mesureur.

We add one hundred grams of sugar measured precisely thanks to a measuring cup.

Captions 10-11, Marie & Jeremy - Le gâteau aux pommes

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Ensuite on mélange grâce à un fouet avec vivacité et énergie.

Then we mix using a whisk with speed and energy.

Captions 14-15, Marie & Jeremy - Le gâteau aux pommes

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Ensuite, grâce à une petite balance de cuisine.

Then, with the help of a small kitchen scale.

Caption 16, Marie & Jeremy - Le gâteau aux pommes

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"Thanks to" is the closest equivalent to grâce à in English. Though Jeremy uses it to talk about inanimate objects, you can just as well use it to refer to a person, someone you're literally thanking:

Merci beaucoup. Grâce à vous, ce mariage, c'était formidable.

Thank you very much. Thanks to you this wedding was great.

Caption 59, Grand Corps Malade - Inch'Allah, en duo avec Reda Taliani

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Grâce has the same Latin root as the Spanish gracias and the Italian grazie, which both mean "thanks." It's also the source of the English word "grace." Like "grace," la grâce (don't forget the circumflex) can mean "elegance," "pardon," and "mercy":

Par lui, tout est grâce et lumière et beauté

Through it, all is grace and light and beauty

Caption 5, Il était une fois - Notre Terre - 9. Les écosystèmes

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La grâce des membres de l'Arche de Zoé pourrait intervenir la semaine prochaine.

The pardoning of the members of Zoe's Ark could occur next week.

Caption 22, Le Journal - L'Arche de Zoé

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Les vénérables vieillards, plusieurs fois centenaires, n'ont pas connu grâce.

The venerable old men, centenarians several times over, did not get any mercy.

Caption 53, Il était une fois... L’Espace - 3. La planète verte

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As you might have guessed, "mercy" is the literal meaning of merci. So when you say "thank you" in French, you're really saying "mercy." And when you say "thanks to" something or someone, you're really saying "grace"!

In "Dimanche soir" (Sunday Night), the slam poet Grand Corps Malade declares his love for his wife in beautiful lines such as: 

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Je l'ai dans la tête comme une mélodie, alors mes envies dansent

I have her in my head like a melody, so my desires dance

Caption 17, Grand Corps Malade - Dimanche soir

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If you didn't see the translation, you might have guessed that envie means "envy." And you would have been right!

Vous ne connaissez que l'envie, la hâte, la rage de les tuer.

You knew only envy, haste, the urge to kill them.

Caption 60, Il était une fois... L’Espace - 3. La planète verte - Part 6

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However, besides désir, envie is also the word for "desire." While un désir is a more general desire, envie connotes yearning, longing, or craving:
 

Il peut rester une envie intellectuelle.

There can remain a mental craving.

Caption 129, Le Figaro - Elle a banni le sucre pendant un an - Part 1

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If you think about it, this double meaning of envie makes a lot of sense, since envy is bound up with desire: if you envy (envier) someone, you covet what they have.

J'envie les caresses

I envy the caresses

Caption 18, Oldelaf - interprète "Bérénice"

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Quitte à en crever de son histoire déçue, de son passé tant envié

Despite wanting to die from her disappointing history, her so envied past

Caption 12, Yaaz - La place des anges

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But envie isn't always so intense. The extremely common expression avoir envie de doesn't mean "to envy" or "yearn for," but simply "to want," "feel like," or "be in the mood for":

Vous avez pas envie de faire la sieste?

You don't feel like taking a nap?

Caption 29, Actu Vingtième - Le Repas des anciens

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J'ai envie d'une limonade.
I'm in the mood for a lemonade.

There's also the expression donner envie (literally, "to give desire"), which means "to make someone want something":

D'avoir des quantités de choses Qui donnent envie d'autres choses

To have things in large quantities That make you want other things

Captions 4-5, Fréro Delavega - Foule Sentimentale

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In English, we have the phrase "green with envy." But in French, one becomes "green with jealousy": vert(e) de jalousie. You can, however, make someone "pale with envy" (faire pâlir d'envie).

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Finally, here's a bizarre quirk of the French language: envie is also the word for "birthmark" and "hangnail." What those have to do with envy and desire is an etymological mystery. 

Pouvoir is an elementary French verb meaning "to be able to." It's an irregular verb, which means it's not conjugated like most other verbs ending in -ir. In this lesson, we'll be focusing on the first-person present indicative form of pouvoir, which has two variants: je peux and je puis (I can). How do we know which one to use?

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Je peux is by far the more common of the two:

Qu'est-ce que je peux faire différemment?

What can I do differently?

Caption 21, Actus Quartier - Manif anti-nucléaire à Bastille

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Puis is actually an archaic conjugation of pouvoir that nowadays is only used in specific, mostly formal contexts. One of them is inversion, when the pronoun and verb switch places:

Que puis-je faire? Puis-je voir ces hommes?

What can I do? May I see these men?

Caption 8, Il était une fois... L’Espace - 3. La planète verte - Part 6

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You would never say que peux-je faire or peux-je voir ces hommes. If you're inverting the first-person present indicative form of pouvoir, you need to use puis. But you could easily rephrase these questions with peux using the constructions est-ce que or qu'est-ce que:

Qu'est-ce que je peux faire? Est-ce que je peux voir ces hommes? 

You're more likely to hear qu'est-ce que je peux or est-ce que je peux than puis-je in everyday speech. Je puis isn't used very often, though it can be found in a few set formal expressions, usually beginning with si:

Si je puis me permettre, essayez ces lunettes

If I may, try these glasses...

Caption 19, Cap 24 - Paris : Alessandro fait les Puces! - Part 2

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C'est un petit peu notre... notre crédo si je puis dire.

It's a little bit like our... our credo, if I may say so.

Caption 18, Télé Lyon Métropole - L'opéra Carmen dans un... boulodrome!

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As a less formal alternative to puis-je (and slightly more formal than je peux), use the conditional form je pourrais:

Alors je pourrais essayer la nuit, Monsieur Watt?

Then I could try at night, Mister Watt?

Caption 2, Il était une fois - Les découvreurs - 13. Stephenson - Part 3

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And don't forget that puis is also an adverb meaning "then":

Puis y en a qui donnent beaucoup moins.

Then there are some who give a lot less. 

Caption 42, Actus Quartier - Repair Café

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