Ni renkontiĝos ĉe la biblioteko post la laboro.

renkontiĝos can be broken down like this:

• renkont- = the idea of meeting
• -iĝ- = a suffix that often means become / get into a state / happen by itself, and it often makes a verb intransitive
• -os = future tense, will

So renkontiĝos means will meet or more specifically will meet up.

In this sentence, with Ni (we), it naturally means we will meet each other / we will meet up.

This is a very common question.

• renkonti usually means to meet someone or to encounter someone
• renkontiĝi usually means to meet up, often with the idea that the people meet one another

So:

• Ni renkontos lin. = We will meet him.
• Ni renkontiĝos. = We will meet up.

If you say Ni renkontos, it can sound more like we will encounter or it may feel as if an object is missing unless the context is very clear.
Ni renkontiĝos is the natural choice when two or more people are arranging to meet each other.

ĉe usually means at, by, with, or in the vicinity of, depending on context.

In ĉe la biblioteko, it means at the library.

This does not necessarily mean physically inside the building. It can mean the library as a meeting point: outside it, near it, or at that location in general.

That is why ĉe is often better than en here.

en la biblioteko means in the library or inside the library.

That is possible if you specifically mean the inside of the building. But ĉe la biblioteko is broader and often more natural for a meeting place.

So the difference is roughly:

• ĉe la biblioteko = at the library
• en la biblioteko = in the library

English often uses at where Esperanto prefers ĉe.

la is the definite article, meaning the.

Esperanto uses la when the thing is specific or known from context. In this sentence, la biblioteko probably means a particular library that both speakers know about.

So:

• ĉe biblioteko would sound odd here
• ĉe la biblioteko = at the library (a specific one)

Esperanto has no indefinite article, so there is no separate word for a/an.

Yes, literally it is after the work, but in natural English we would usually say after work.

Esperanto often uses la in places where English does not. Here la laboro refers to the work period understood from context, such as the workday, the shift, or the work we have today.

So post la laboro is a normal way to say after work.

You may also sometimes see post laboro, which can sound a bit more general, but post la laboro is very natural when both people know which work period is meant.

Because both nouns are objects of prepositions:

• ĉe la biblioteko
• post la laboro

After a preposition, Esperanto normally does not use the accusative -n.

So:

• ĉe la biblioteko = correct
• post la laboro = correct

The accusative is mainly used for a direct object or sometimes for direction/motion in certain expressions, but not here.

The word order is flexible.

This sentence uses a very straightforward order:

• Ni = subject
• renkontiĝos = verb
• ĉe la biblioteko = place
• post la laboro = time

But you could also say:

• Post la laboro ni renkontiĝos ĉe la biblioteko.
• Ĉe la biblioteko ni renkontiĝos post la laboro.

All of these are grammatical. The difference is mainly emphasis.

Putting Post la laboro first emphasizes the time.
Putting Ĉe la biblioteko first emphasizes the place.

A few useful pronunciation points:

• ĉ is pronounced like ch in church
• ĝ is pronounced like j in judge
• Stress in Esperanto is always on the second-to-last syllable

So:

• ĉe sounds roughly like cheh
• renkontiĝos is pronounced roughly ren-kon-TEE-jos, with the stress on ti

That stress pattern is one of the most regular and helpful features of Esperanto.