In German, there are two words that can be translated as "the accident": der Unfall, which is when you fall off your bicycle, and der Zufall, which refers to a random occurrence or coincidence. When you talk about something happening zufällig ("accidentally"), it implies this aspect of randomness. When we want to talk about something happening "accidentally" simply in the sense of it being "unintentional," there is a better adverbial phrase to use:

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Jetzt bin ich aus Versehen zu weit gelaufen und muss den Bus nach Hause nehmen.

Now I accidentally walked too far and have to take the bus home.

Caption 38, Shuah - Auf der Straße in Berlin

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A less commonly used synonym for aus Versehen is ohne Absicht. Die Absicht means "the intention," "the aim," or "the purpose." 

Das war Absicht.

That was on purpose.

Caption 16, JoNaLu - Überall Banditen

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When we speak about something done "intentionally" or "on purpose," we can use either mit Absicht or the adverb absichtlich. 

Ich konnte ja nicht wissen, dass du Max mit Absicht belogen hast.

I couldn't indeed have known that you lied to Max on purpose.

Caption 25, Die Pfefferkörner - Cybermobbing - Part 6

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Jemand hat Jannik Sternberg absichtlich vom Gerüst geschubst.

Someone intentionally shoved Jannik Sternberg off the scaffolding.

Caption 50, Großstadtrevier - Von Monstern und Mördern - Part 2

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Further Learning
Look absichtlich and aus Versehen up on Yabla German. Think of a few scenarios in which you would need to clarify whether an action was intentional or not and build a few sentences in the past tense. Here are a few pairs of nouns and verbs to get you started:
die Mail / weiterleiten
dich / anrufen
seinen Kaffee / trinken
die Tasche / zu hause lassen

A syllabic abbreviation is formed by taking the first syllable of several words and putting them together. It is a common practice in German that is rarely used in English. One English syllabic abbreviation you may recognize is Interpol, formed from "international police." You may not have known that the word for the Nazi secret police terror organization the Gestapo comes from a syllabic abbreviation of die Geheime Staatspolizei. Note that the gender of a German abbreviation is determined by the main noun of the word it is derived from, hence die Gestapo.  Let's take a brief look at some other common German syllabic abbreviations.

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Stasi-Akten aus dem Schredder...

Stasi [acronym for state security] files from the shredder...

Caption 2, DDR zum Anfassen - Ganz tief im Westen

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The notorious East German secret police die Stasi also had their own syllabic abbreviation, in this case formed from die Staatssicherheit, the Ministry of State Security.
 

Das sind Fragen für die Kripo [Kriminalpolizei],

Those are questions for the criminal investigation department,

nicht für uns.

not for us.

Caption 14, Großstadtrevier - Von Monstern und Mördern

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In this case, die Kripo is derived from die Kriminalpolizei, the department of the police that specializes in criminal investigations. But let's get away from the police while we still can! 
 

Du musst in die Kita [Kindertagesstätte].

You have to go to daycare.

Caption 20, Mama arbeitet wieder - Kapitel 1: Alle haben sich lieb

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Ironic, considering its German origin, that the most common German word for a daycare center is not der Kindergarten, but rather die Kita, an abbreviation of die Kindertagesstätte.
 

Ich stehe hier vor dem Audimax.

I am standing here in front of the main lecture hall.

Caption 28, Universität - Karlsruhe

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At the university level, the term das Audimax is standard student parlance for the largest lecture hall of a given university, and in this case is an abbreviation of the Latin term auditorium maximum.

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Further Learning
Take a look at this German article about syllabic abbreviations and go to German Yabla to find some of the words and phrases used in context.  

At Yabla, we like to write out words in their entirety (for instance, 2016 is zweitausendsechzehn) in our video captions so that you can learn them better. But you may also come across abbreviations and acronyms in German articles that you are reading, so it's not a bad idea to catch up on some of the more common ones. (An acronym is an abbreviation too, but it's used in speaking.) You won't find many actual abbreviations in Yabla videos since they aren't usually spoken, but there are plenty of examples where words or phrases would be abbreviated if they were in standard written form! 

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Unterschrift, Stempel und so weiter und so weiter, ne?

Signature, stamp, and so on and so forth, right?

Caption 23, Großstadtrevier - Von Monstern und Mördern

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One of the most common abbreviations of all is usw., which is short for und so weiter, and pretty much interchangeable (even in German) with "etc."
 

Merkel hat noch TÜV [Technischer Überwachungsverein]

Merkel still has a TÜV [technical inspection organization certificate]

bis zweitausenddreizehn.

that's valid until two thousand thirteen.

Captions 32-33, Der Merkelpilot - der kleine Mann, der es macht

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If your car won't pass the TÜV (an acronym, with the "Ü" pronounced like the "oo" in "foot") inspection in Germany, you'd better get it fixed or get rid of it, because you can't drive it on the street without the certificate from the Technical Inspection Organization or Technischer Überwachungs-Verein. This non-profit organization (eingetragener Verein or e. V.), responsible for overseeing technical inspection procedures in Germany, even owns a registered trademark on the term. Hopefully Chancellor Merkel will get her TÜV approved this year! 
 

Wir sind sehr zuversichtlich beziehungsweise sehr froh auch.

We are very confident, or rather, very happy too.

Caption 26, Strothoff International School - Interview mit dem Rektor

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The word beziehungsweise is often shortened to bzw. in informal writing, and is one of the most commonly abbreviated German words. It's also a very effective word that has no single-word equivalent in English, meaning "and/or" all in a single word. It's a bit tricky to translate because "and/or" is just not an elegant solution, so it's often translated as "respectively" or "alternately."

The undisputed champion of German acronyms, however, must be "for example": 
 

Hier haben wir zum Beispiel Rohkaffee aus Kolumbien.

Here we have, for example, raw coffee from Colombia.

Caption 8, Kaffee - Noch von Hand gemacht

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"For example" or zum Beispiel takes the acronym z. B., and this being German, don't forget to capitalize the "B" even in its short form! 

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Further Learning
Take a look at this list of common German abbreviations and go to German Yabla to find some of the words and phrases used in context.