Magister auctionis auctionem hereditariam parat.

Because magister is the subject of the sentence: it is the person who is doing the action.

• magister = the master / manager / head
• nominative singular, masculine

Since parat means prepares, magister is the one who prepares.

auctionis is genitive singular.

It depends on magister and shows a relationship like of the auction:

• magister auctionis = the master/manager of the auction

A native English speaker can think of this as a common Latin pattern:

• noun + genitive
• X of Y

So here:

• magister = master/manager
• auctionis = of the auction

The base noun is auctio, auctionis (feminine), a 3rd-declension noun.

Because they are in different cases and do different jobs in the sentence.

They both come from auctio, auctionis:

• auctionis = genitive singular → of the auction
• auctionem = accusative singular → the direct object

So the sentence uses the same noun twice, but differently:

• magister auctionis = the manager of the auction
• auctionem hereditariam = a hereditary/inherited auction, the thing being prepared

Latin changes noun endings depending on function, much more than English does.

This phrase is the direct object of parat.

It tells you what the subject is preparing:

• parat = prepares
• auctionem hereditariam = the hereditary/inherited auction

Both words are in the accusative singular feminine:

• auctionem = accusative singular of auctio
• hereditariam = accusative singular feminine of hereditarius

That matching is important: the adjective hereditariam agrees with auctionem.

Because it is an adjective describing auctionem, and Latin adjectives must agree with the nouns they modify in:

• gender
• number
• case

Here:

• auctionem is feminine singular accusative
• so hereditariam must also be feminine singular accusative

That is why it has the ending -am.

Grammatically, it is simply the adjective from hereditarius. Depending on context, it can be understood as:

• hereditary
• of an inheritance
• inherited
• connected with an estate

In a sentence like this, English may choose the most natural wording rather than a perfectly literal one. The important Latin point is that hereditariam modifies auctionem.

parat is:

• present tense
• 3rd person singular
• active voice
• from paro, parare = to prepare

So it means:

• he prepares
• she prepares
• it prepares

Since the subject is magister, we understand it as he or simply the manager prepares.

Latin normally has no articles.

So magister can mean:

• the manager
• a manager

And auctionem can mean:

• the auction
• an auction

English has to choose one based on context, but Latin usually leaves that unstated.

Yes, it could be rearranged, because Latin word order is more flexible than English word order.

The basic sentence is:

• Magister auctionis auctionem hereditariam parat.

But Latin could also say things like:

• Auctionem hereditariam magister auctionis parat.
• Parat magister auctionis auctionem hereditariam.

The endings show the grammatical roles:

• magister = subject
• auctionis = genitive
• auctionem hereditariam = object
• parat = verb

So word order often affects emphasis more than basic meaning.

Because the phrase means of the auction, referring to one auction.

If it were plural, it would mean something like of the auctions. The singular genitive auctionis shows a single auction.

So:

• magister auctionis = manager of the auction
• not manager of several auctions

At least, that is what the grammar suggests.

You would look them up like this:

• magister, magistri — masculine noun, teacher / master / manager
• auctio, auctionis — feminine noun, auction
• hereditarius, -a, -um — adjective, hereditary / inherited
• paro, parare, paravi, paratus — verb, prepare

This is useful because the forms in the sentence are not always the same as the dictionary form:

• auctionis and auctionem both come from auctio
• hereditariam comes from hereditarius
• parat comes from paro

A helpful grammatical breakdown is:

• Magister — nominative singular masculine, the subject
• auctionis — genitive singular, of the auction
• auctionem — accusative singular feminine, the direct object
• hereditariam — adjective agreeing with auctionem
• parat — 3rd person singular present, prepares

So structurally it is:

• [Subject] [genitive dependent on subject] [object + adjective] [verb]

That makes it a very useful sentence for seeing how Latin endings show relationships clearly.