There are two ways of saying "either... or..." in French, and they both involve repeating one word. The first is the construction soit... soit.... Soit is a conjunction that marks a set of alternatives, and it is also spelled the same as the third-person present subjunctive form of the verb être (to be):  

Les médecins étaient soit morts, soit partis.

The doctors were either dead or gone.

Caption 4, TV8 Mont Blanc - De retour de Haïti

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BANNER PLACEHOLDER

A similar construction with soit is que ce soit... que ce soit..., which can best be translated as "be it... or...":  

Que ce soit déposer dans le sable,

Be it landing on sand,

que ce soit déposer dans la neige...

or on snow...

Caption 26, Le Journal - École de pilotage

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The second way of saying "either... or..." is ou... ou.... Ou by itself just means "or" (not to be confused with où, "where"), but when it is repeated to describe two or more choices or alternatives, the first ou means "either":

Ou vous pouvez le laisser tout simplement sur la plage,

You could either simply leave it on the beach

ou vous en servir comme cendrier.

or you could use it as an ashtray.

Caption 15, Jean-Marc - La plage

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Sometimes, bien can be added to ou to emphasize the distinction:

Ou bien il est très heureux, ou bien il est misérable. 

Either he's very happy or he's miserable.    

Note that you will often see a comma separating the alternatives soit... soit... and ou... ou... (soit morts, soit partis). 

Now that we've learned how to say "either... or...," we'll move on to its opposite, "neither... nor...." There is only one way to say this in French: ni... ni....

Ni vu ni connu

Neither seen nor known [on the sly]

When using ni... ni... with verb phrases, add a ne in front of the verb:

Nous ne sommes ni les premiers, ni les derniers.

We are neither the first nor the last.

Caption 3, La Conspiration d'Orion - Conspiration 2/4

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Sometimes, you might just find a single ni:

Cette femme habite un monde sans foi ni loi...

This woman inhabits a world without faith or law...

Caption 19, Le Journal - Milady

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BANNER PLACEHOLDER

So now, if you're ever asked to recite the unofficial creed of the US Postal Service in French, you won't hesitate to say:

"Ni la neige, ni la pluie, ni la chaleur, ni la nuit n'empêchent de fournir leur carrière avec toute la célérité possible".

"Neither snow, nor rain, nor heat, nor gloom of night stays these couriers from the swift completion of their appointed rounds." 

(The creed is actually a line from Herodotus.)