A number of German verbs have legen as a root. Let's go alphabetically and explore some of the most common words. 

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The verbs auflegen and einlegen each have a number of different translations. The verb auflegen describes putting something on top of something else and, by extension, can mean "to hang up the phone" or "to play music as a DJ." It is also used to refer to publishing an edition. The verb einlegen literally means "to put something into something," which can be physical in the case of fermenting or pickling vegetables (eingelegtes Gemüse) or immaterial in the case of raising an objection (Einspruch einlegen) or taking a break (eine Pause einlegen). 

Max? -Henri, ich muss jetzt leider auflegen.

Max? -Henri, I have to hang up now, unfortunately.

Caption 28, Die Pfefferkörner: Endspurt

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Erschöpft legen Franz Obermeier, seine Söhne und Helfer eine Pause ein.

Exhausted, Franz Obermeier, his sons and helpers take a break.

Caption 8, Die letzten Paradiese Schätze der Natur: Südtirol

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The verb festlegen is more straightforward, as it is always related to determining, defining, or ascertaining something. 

Figuren und Reitstil sind seit Jahrhunderten genau festgelegt.

The figures and riding style have been precisely defined for centuries.

Caption 46, Die Alpen von oben - S1 E2: Von Graz ins Ausseerland

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And sich hinlegen simply means "to lie down." 

Es legte sich hin und schlief sofort ein.

She laid herself down and immediately fell asleep.

Caption 43, Märchen - Sagenhaft: Goldlöckchen und die drei Bären

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You'll hear the verb loslegen quite often at the beginning of Yabla videos: 

Ja, ja, das geht dann auch schneller. -Na, dann lass uns loslegen!

Yes, yes, then it will also go faster. -Well, then let's get started!

Caption 44, Abenteuer Nordsee: Unter Riesenhaien und Tintenfischen

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The verb offenlegen means "to reveal."

Das Null-zu-null beim leidenschaftlich kämpfenden Hamburger SV legte das Problem des Rekordmeisters offen.

The zero to zero outcome against a Hamburg squad, [ho fought passionately, revealed the master record champion's main problem.

Captions 19-20, Fußball pur Paderborn: im Rausch

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The first non-separable verb of our list is überlegen, "to consider."  

Schön... Lass mich mal überlegen.

Nice... Let me think about it.

Caption 28, Deutsch mit Eylin: Anruf bei einer neuen Bekannten

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The verb verlegen means "to misplace" or "to lose," but it is also an adjective that means "embarrassed." 

„Ach so, hm“, meinte Frederick verlegen. „Dann können wir ja weitergehen.“

"Ah, so, hm," said Frederick sheepishly. "Then, indeed, we can keep walking."

Caption 34, Piggeldy und Frederick: Das Lachen

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The verb zurücklegen means "to put back," but also "to cover" in the context below:

Elftausend Kilometer haben Sie mit dem Fahrrad zurückgelegt.

You have traveled eleven thousand kilometers [nearly seven thousand miles] with the bicycle.

Caption 8, Mountainbiken: Auf dem Anden-Trail

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Further Learning
You'll find many videos with these verbs used in context on Yabla German. You can also do a search for many other verbs with the root legen, such as ablegen, anlegen, beilegen, freilegen, and more. 

Using the German adjective verlegen and the verb verlegen could lead to some embarrassing mix-ups if their meanings are not clearly understood. As you can see in this week's premiere episode of a new season of Die Pfefferkörner, the meaning of the adjective verlegen is... embarrassed!

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Um eine Ausrede bist du ja nie verlegen.

You are never embarrassed to make an excuse.

Caption 6, Die Pfefferkörner - Gerüchteküche

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The German verlegen is often also translated to English as shy, awkward, bashful, or as you see in its adverbial form in this video on Yabla:

„Ach so, hm“, meinte Frederick verlegen.

"Ah, so, hm," said Frederick sheepishly.

Caption 34, Piggeldy und Frederick - Das Lachen

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On the other hand, the verb verlegen has to do with more concrete matters:

... um eine ganz normale Hartsteinbetonplatte, die wir verlegen.

... with a totally normal hard stone concrete panel that we are installing.

Caption 19, Schadstoffarme Straßen - Neue Gehwegplatten für reinere Luft

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Hundert Jahre nachdem das berühmte Kinderbuch erschienen ist,

A hundred years after the famous children's book was published,

hat der Regisseur die Story in den Zweiten Weltkrieg verlegt.

the director has relocated the story to the Second World War.

Captions 35-37, Kinotipp - Battleship und Unter Wölfen

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The verb verlegen can also be translated as to publish, postpone, or evacuate. The German word for "publisher," which you see in nearly every German book, is der Verlag. Here you can easily see the connection to the verb verlegen.

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Further Learning
For some advanced learning on the topic, go to the online Wiktionary and see some other examples of verlegen in context and some other related words.