The separable verbs anheben and aufheben look very similar and have meanings related to the base verb heben, which is usually translated as "to lift," or "to raise," and is the Germanic root of the English verb "to heave."

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Let's first take a look at anheben: 

Heb einfach den Riegel an und komm herein!
Simply lift the latch and come inside!

Caption 48, Märchen - Sagenhaft - Rotkäppchen und der Wolf

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Die Hüfte hebt der Springer an,

The jumper lifts his hips

indem er die Hacken Richtung eigenes Kreuz drückt.

by pressing his heels toward his own lower back.

Captions 29-30, Olympische Spiele - Hochsprung

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Dazu hebst du ein Ski-Ende an und versetzt es auf die Seite.
For this, you'll lift the back of the ski and shift it to the side.

Caption 14, Skifahren lernen - Erste Vorübungen im flachen Gelände

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Dieses Bein anheben. Ah, ja, genau.
Lift that leg. Oh, yes, exactly.

Caption 17, TEDx - Lebenslange Fitness

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Anheben is used when you want something to be lifted just a little or for a short amount of time, such as asking somebody to lift up the sofa so you can vacuum under it.

Let's contrast that now with aufheben:

Oma, kann ich die Münze aufheben? 
Grandma, can I pick up the coin?

Caption 4, Ivana erzählt Witze - Fritzle und die Oma

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Fritzle, heb mich auf.
Little Fritz, pick me up.

Caption 14, Ivana erzählt Witze - Fritzle und die Oma

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Aufheben is usually translated as "to pick up" but can also mean "to save" in the sense of aufbewahren. It is also used in an idiomatic expression that can be useful: 

Ich hebe es mir für einen anderen Tag auf.
I'll save it for another day.

Caption 12, Hausputz - mit Eva

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Further Learning
In a forthcoming lesson we'll explore the differences between abheben and hochheben, yet more variations of the root verb heben. Das hebe ich mir aber für einen späteren Newsletter auf! 

You are likely already very familiar with the verb kaufen and its antonym verkaufen: 

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Die Leute gehen in den Supermarkt, kaufen etwas und wissen gar nicht, woher es kommt.

People go to the supermarket, buy something and don't even know where it comes from.

Captions 12-13, Bundesregierung - Der Tomatenfisch

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Wir verkaufen Brezeln mit Schokolade.

We sell pretzels with chocolate.

Caption 20, Berlin - Judith und die „Brezel Bar“

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You probably know the verb einkaufen as well: 

Fritzle geht mit seiner Oma einkaufen.

Little Fritz goes shopping with his grandma.

Caption 2, Ivana erzählt Witze - Fritzle und die Oma

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You may, however, be less familiar with the separable verb ankaufen. It is very similar to kaufen ("to buy"), but has a very specific context related to a person or a company that offers to buy items on a regular basis: 

Der Juwelier kauft Gold an.
The jeweler buys gold. 

Der Autohändler kauft Gebrauchtwagen an.
The car dealer buys used cars. 

The separable verb aufkaufen can be used to mean "to buy up" or to buy in large quantities:

Ursprünglich hat er alte Anzüge aufgekauft, so aus den Zwanzigern und so.

Originally, he bought up old suits from the twenties and such.

Caption 6, Jonathan Johnson - Herr von Eden

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The verb aufkaufen is also used when a company takes over or acquires another company: 

Facebook und Amazon wurden von Google aufgekauft.
Facebook and Amazon were acquired by Google.

Please note that the above statement is purely hypothetical and is in no way intended to represent the intention of any companies, real or imagined! 

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Further Learning
Read this excellent article on the four verbs above, and see this list of words that use the root kaufen. You may then go to Yabla German and find examples of the words used in a real world context.

The most typical New Year’s greeting in Germany is the slang phrase Einen guten Rutsch ins neue Jahr! or simply Guten Rutsch! These can be translated literally as “A good slide into the New Year!” or “Good slide!”

Dann wünsch' ich euch schöne Weihnachten,

In which case I wish you [plural] a wonderful Christmas,

'nen [einen] guten Rutsch ins neue Jahr und viel Spaß beim Anschauen. Tschüß!

a good slide to the new year and a lot of fun while watching. Bye!

Captions 84-85, Weihnachtsinterviews - Cettina in Linkenheim

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The word Rutsch means a downward sliding movement, and the greetings are meant to convey a smooth transition into the New Year, though the origin of the phrase is still uncertain. Let’s have a look at the more ordinary ways Rutsch and rutschen are used. 

In an interview, Germany’s top windsurf star, Bernd Flessner, is asked if he can get from Altenteil Halpen to the Lighthouse in einem Rutsch durch (in one go [literally “through in one slide”]). The answer? Yes!

In einem Rutsch durch? -In einem Rutsch.

In one go? [literally “through in one slide”] -In one go.

Caption 9, Das "Race around Fehmarn" - Neuer Surfrekord

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A professor involves his talking pet bird named Dodo in demonstrating Newton's law of inertia using the “Dodomobil," (really just a shoebox). The verb rutschen (to slide or skid) is in more common usage than the noun Rutsch:

Dann... theoretisch, fährt dann das Dodomobil

Then… theoretically, the Dodomobile then continues to drive...

oder rutscht, besser gesagt...

or slide, rather…

rutscht das Dodomobil unendlich weiter.

the Dodomobile continues to slide infinitely.

Captions 38-39, Das 1. Newtonsche Gesetz - erklärt am Beispiel des Dodomobils

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Piggeldy and Frederick are playing on a Rutschbahn (a slide):

„Richtig", schrie Frederick und rutschte die Rutschbahn hinunter.

"Right," shouted Frederick and slid down the slide.

Caption 36, Piggeldy und Frederick - Spielen

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Watch out, Grandma! Here rutschen takes on another meaning:

Dann rutscht die Oma auf einer Bananenschale aus.

Then the grandmother slips on a banana peel.

Caption 13, Ivana erzählt Witze - Fritzle und die Oma

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So whether you are sliding down a slide, sliding into the New Year, or slipping on a banana peel, understanding this versatile word can help you have a guten Rutsch ins Deutsch!