The words in the above headline all mean "at least," but it can be confusing as to which context is correct for the right word. Note too that "at least" is a prepositional phrase in English, whereas in German the expression is usually a simple adverb. The adverb mindestens is probably the most common: 

In Deutschland ist es so:

In Germany, it's like this:

Asylbewerber müssen mindestens drei Monate warten.

Asylum applicants must wait at least three months.

Caption 35, Flüchtlingskrise - 10 Vorurteile, die nicht stimmen

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The adverb mindestens usually, as in the first case above, refers to a length of time or an amount of something:

Heutiger Hochwasserstand: wieder mindestens zehn Zentimeter.

Today's flood water level: at least ten centimeters again.

Caption 44, Die Klasse - Berlin '61

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It's used somewhat less often to refer to circumstances: 

Zwar haben die Zuwanderer in der Regel eine hohe

Indeed, the immigrants normally have a high

oder auch mittlere Qualifikation,

or also mid-level qualification

die also mindestens einem deutschen, äh, Abschluss.

that is at least on par with a German, uh, degree.

Captions 35-36, Rhein-Main-TV aktuell - Mehr Beschäftigung in Rhein-Main

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The related adverb zumindest, on the other hand, is used much more commonly for situations rather than lengths of time or amounts: 

Nun, zumindest habe ich meinen ersten Anruf hinter mir.

So, at least I have my first call behind me.

Caption 69, Berufsleben - das Vorstellungsgespräch

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Be careful not to jumble mindestens and zumindest into one (incorrect) word, something that even native German speakers occasionally do!

Eislaufen ist leicht,

Ice skating is easy,

zumindestens [sic, zumindest] leichter als auf Vanessas Party eingeladen zu werden.

at least easier than getting invited to Vanessa's party.

Captions 82-83, Küss mich, Frosch - Leb wohl, kleiner Prinz

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So remember that mindestens and zumindest are real words, either of which would have been correct in the above sentence, but "zumindestens" is not a proper word at all!

Even less common, but making things even more complex, is the adjective min­des­te / min­des­ter / min­des­tes. This adjective can also be nominalized, or turned into a noun, such as das Mindeste (the least thing), or zum Mindesten (at least). 

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Further Learning
Go to Yabla German and find examples of the above adverbs and adjectives to see how they are used in a real-world conversational context. To go even deeper into the adjectival usage, read the Duden page for min­des­te / min­des­ter / min­des­tes.  

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