Its forms

The accusative is again nothing more than a set of article endings and a few personal pronouns. Those are:

ARTIKEL
den (m)
das (n)
die (f)
die (pl)

PERSONALPRONOMEN
mich = me
dich = you (singular)
ihn = him
es = it
sie = her
uns = us
euch = you (plural)
sie/Sie = they or you (sir/madam)(formal)

You can easily memorize its article endings by remembering that the only difference to the nominative is found in the masculine form which ends in n. The n has two strokes down and the accusative is the second case of four.

When do I use it?

There are four situations in which you need to use the accusative:

1. After an accusative preposition
2. After a two-way preposition (will be explained later)
3. After an accusative verb
4. For the gift if there are two objects in one sentence (direct & Indirect objects)

1. After an accusative preposition

A preposition stands as its name already gives it away, pre, i.e., before a noun and usually gives us information, e.g., about location or time, but, at times, also about a mode, e.g., movement. There are a few prepositions that always require the accusative:

entlang along
durch through
um around/at
gegen against
für for
bis until
ohne without

After any of these prepositions, you will have to use the accusative. You‘ll never come across a durch der or ohne der etc.

2. After a two-way-preposition answering "Wohin?"

These are the following:

vor before/in front of
hinter behind
über above
unter under
an at/on
zwischen in between
auf on (top)
in in(to)
neben next to

Wohin fahren wir? In die Türkei. (accusative)
Where-to go we? Into the Turkey.

There will be an extra chapter about the two-way prepositions later with a precise description of their usage.

3. After an accusative verb

The accusative is used for the direct object of a sentence:

Der Hund frisst einen Knochen.
The dog eats a bone.

The dog is doing the eating, so it is the subject of that sentence. What‘s left is the bone. As there can be only one subject per sentence, the bone must be the object.

An object usually is the thing or person to which the action is directed. The object of the action is not always that obvious, but you can usually assume that any other (pro)noun in a sentence besides the subject is an object. And as only the subject allows the use of the nominative, the object can not use the nominative case again.

And although an object could be used in the accusative, the dative or the genitive, in 90% of the situations you will have to use the accusative. A good rule of thumb is always to use the accusative (for an object) in case of doubt. Chances are high that you will guess correctly. To raise your chances it suffices to learn when to use the dative for an object which I'll talk about a bit below and more later on. The genitive is insignificant and will be ignored throughout this course. In B2 you'll take a closer look at the genitive but for now, just acknowledge its existence and move on.

4. For the "gift" if there are two objects

There are verbs that can take two objects. In almost all the cases you will then have one dative and one accusative object. There are only three exceptions for now. * The good news is that these verbs with two objects are universal. That means, even on Mars, they would have such verbs. One example:

Bringst Du mir einen Wein?
Will you bring me (a bottle of) wine?

You could leave out the mir but not the Wein. In all languages you can bring something to somebody. And the thing that is being „given“ to someone, the gift, is always used in the accusative. Some more common verbs of this kind are:

bringen to bring
geben to give
kaufen to buy
machen to make
schenken to give as a gift
erzählen to tell
sagen to say
verraten to give away (e.g. a secret)
leihen to borrow/lend

How do I find the accusative?

You can find the accusative by asking the questions wen for persons or was for non-persons.

Die Frau liebt ihren Mann. Wen liebt die Frau? Ihren Mann.
The woman loves her husband.

Der Chef kauft einen neuen Computer. Was kauft der Chef? Einen neuen Computer. The boss is buying a new computer.

One trick to remember the questions:
wen ends in n just like den or einen (accusative masculine)
was ends in s just like das or dieses (accusative neuter)

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*those are lehren > to teach, kosten > to cost and nennen > to name which use two accusatives and are very rarely used so that we can calmly ignore them.