Procul in campo cervus celer currit, dum cerva inter arbores paulum manet.

A quick parse looks like this:

• Procul — adverb, far off / at a distance
• in campo — prepositional phrase, in the field
  • campo is ablative singular
• cervus — nominative singular, the subject of currit
• celer — adjective agreeing with cervus, swift
• currit — 3rd person singular present, runs
• dum — conjunction, here while
• cerva — nominative singular, the subject of manet
• inter arbores — prepositional phrase, among the trees
  • arbores is accusative plural
• paulum — adverbial accusative, a little / for a little while
• manet — 3rd person singular present, remains / stays

So the sentence has two main parts:

• cervus celer currit — the swift deer runs
• dum cerva inter arbores paulum manet — while the doe stays a little among the trees

Because in uses different cases depending on meaning:

• in + ablative = in / on a place, with no motion into it
• in + accusative = into a place, showing motion toward it

So:

• in campo = in the field
• in campum = into the field

Here the deer is already being described as located in the field, so Latin uses the ablative: campo.

Procul and in campo are doing different jobs:

• procul tells you the place is far away from the speaker or observer
• in campo tells you where it is: in the field

So together they mean something like:

• Far off, in the field...

English often uses both kinds of information too: one word gives distance from us, the other gives location in itself.

Because celer is an adjective, while celeriter is an adverb.

• celer = swift, fast
  It describes the deer
• celeriter = swiftly, quickly
  It would describe the running

So:

• cervus celer currit = the swift deer runs
• cervus celeriter currit = the deer runs quickly

Latin can express either idea, but they are not exactly the same.

Latin word order is much freer than English word order.

In English, word order usually tells you what belongs together. In Latin, the endings do most of that work. Cervus and celer match in:

• gender: masculine
• number: singular
• case: nominative

That agreement shows that celer modifies cervus, no matter whether it comes before or after.

So all of these are possible Latin patterns:

• celer cervus
• cervus celer

The exact word order may reflect style or emphasis, but both are grammatically fine.

Here dum means while.

It introduces an action happening at the same time as the main action:

• cervus... currit — the deer runs
• dum cerva... manet — while the doe stays

So the sense is simultaneous action.

But dum does not always mean while. In other contexts, it can also mean things like:

• until
• sometimes provided that, in certain constructions

So you always need to check the context. In this sentence, while is the natural meaning.

Because the sentence is describing two actions as happening at the same time:

• currit — runs
• manet — stays / remains

Both are 3rd person singular present indicative.

With dum meaning while, Latin often uses the present tense to show an action going on at the same time as another action. English often does something similar, though English may choose either a simple present in translation or a more natural continuous form depending on context.

So the idea is roughly:

• The deer runs, while the doe stays...
• or more naturally in English, The deer is running, while the doe stays / is staying...

Because inter takes the accusative case.

So:

• inter arbores = among the trees / between the trees

Here arbores is accusative plural.

This is something you simply learn with the preposition:

• inter + accusative

Even though the meaning is about location, Latin prepositions do not always work the way English ones do. The important thing is the case that the preposition governs.

Here paulum means a little or for a little while.

In this sentence it works adverbially, modifying manet:

• paulum manet = she stays a little / she remains for a short time

This is a common Latin feature: a neuter singular form can be used adverbially.

So paulum is not agreeing with cerva. It is not an adjective describing the doe. It is giving the extent or duration of the action.

Because Classical Latin has no articles.

Latin does not have separate words that directly equal English:

• the
• a / an

So cervus can mean, depending on context:

• a deer
• the deer

And cerva can mean:

• a doe
• the doe

English translations have to choose whichever sounds best in context.

They are related nouns with different natural gender:

• cervus = male deer, often stag or hart
• cerva = female deer, often doe or hind

So this sentence is contrasting a male and a female deer.

This is not just a random ending change like English adding a pronoun. These are standard Latin noun forms:

• cervus — 2nd declension masculine
• cerva — 1st declension feminine

Because Latin usually does not need them.

The verb endings already tell you the person and number:

• currit = he/she/it runs
• manet = he/she/it remains

And the nouns cervus and cerva already name the subjects clearly. So adding pronouns would be unnecessary unless Latin wanted special emphasis or contrast.

This is very normal in Latin. It is often called a pro-drop language: the subject pronoun can be left out because the verb ending gives enough information.