Vor dem Einzug packen wir Bücher und Kleidung in jeden Umzugskarton.

It means before the move-in or more naturally before moving in.

• vor = before
• der Einzug = the move-in, moving into a new home/apartment

So Vor dem Einzug is a time expression telling you when the packing happens.

Because vor takes the dative case when it is used in a time expression.

Here, vor dem Einzug means before the move-in, so:

• der Einzug is a masculine noun
• masculine dative singular = dem Einzug

So:

• nominative: der Einzug
• dative: dem Einzug

This is a basic German main-clause rule: the finite verb must come in second position.

The sentence begins with the time phrase Vor dem Einzug, so that takes the first position. Then the verb must come next:

• Vor dem Einzug = position 1
• packen = position 2
• wir = after the verb

So German says:

• Vor dem Einzug packen wir ...

not

• Vor dem Einzug wir packen ...

This is very common in German.

Because once something else is placed at the beginning of the sentence, the verb still has to stay in second position.

Compare:

• Wir packen Bücher und Kleidung ... = neutral word order
• Vor dem Einzug packen wir Bücher und Kleidung ... = the time phrase is moved to the front

So the subject wir moves after the verb.

German often leaves out articles when talking about things in a general sense.

Here:

• Bücher = books
• Kleidung = clothing/clothes

So Bücher und Kleidung means books and clothing, not necessarily the books and the clothing.

A few details:

• Bücher is a plural count noun
• Kleidung is usually treated as an uncountable noun, like clothing in English

That is why bare nouns sound normal here.

They are the direct object of packen, so they are in the accusative case.

However, in this sentence you do not see a special ending:

• Bücher looks the same in nominative and accusative plural
• Kleidung also looks the same in nominative and accusative singular

So they are accusative by function, even though their forms do not visibly change.

Because in can take either the accusative or the dative, depending on meaning.

Here it means movement into something:

• packen ... in jeden Umzugskarton = pack ... into every moving box

That idea of direction/change of location uses the accusative.

So:

• jeden Umzugskarton = accusative masculine singular

If it were about location inside a box, not movement into it, you would use the dative:

• in jedem Umzugskarton = in every moving box

So the contrast is:

• accusative: movement into
• dative: location in

Because Umzugskarton is a masculine noun, and after in with a directional meaning, it is in the accusative singular.

The adjective-like word jeder changes its ending depending on case, gender, and number.

Here the pattern is:

• nominative masculine: jeder
• accusative masculine: jeden

So:

• in jeden Umzugskarton

is grammatically correct.

Umzugskarton means moving box or cardboard box for moving.

It is a German compound noun:

• der Umzug = move / relocation
• der Karton = box / carton

Together:

• Umzugskarton = moving box

German very often builds long nouns by combining smaller nouns like this.

Because jeden means each/every, and in both German and English that is usually followed by a singular noun.

So:

• jeden Umzugskarton = each/every moving box

German often uses this distributive singular to talk about all items one by one.

A plural version would need different wording, for example:

• in alle Umzugskartons = into all the moving boxes

That is slightly different in nuance from into every moving box.

A helpful breakdown is:

• Vor dem Einzug = time expression
• packen = verb
• wir = subject
• Bücher und Kleidung = direct object
• in jeden Umzugskarton = prepositional phrase showing destination

So the pattern is:

Time expression + verb + subject + object + destination

This is a very common German sentence pattern when something other than the subject comes first.