Mi aliĝas; la bibliotekisto registras mian membrecon ĉe la giĉeto.

The semicolon separates two complete clauses that are closely connected.

• Mi aliĝas is one full statement
• la bibliotekisto registras mian membrecon ĉe la giĉeto is another full statement

The semicolon shows that these two actions belong together in the same situation, but they are still separate actions with separate subjects.

A period would also be possible. A comma would not be correct here.

Both are in the present tense because they end in -as.

In Esperanto:

• -as = present
• -is = past
• -os = future
• -us = conditional
• -u = command or wish
• -i = infinitive

So:

• aliĝas = is joining, joins
• registras = is registering, registers

As often happens in Esperanto, the present tense can describe either something happening now or a general routine, depending on context.

Aliĝas is the present-tense form of aliĝi.

This verb means something like to join, to sign up, or to enroll.

A useful thing to notice is the element -iĝ-, which often gives the idea of becoming or entering a state. So aliĝi has the sense of becoming affiliated or becoming part of something.

That is why Mi aliĝas works well for I’m joining or I’m signing up.

Because aliĝi is normally used intransitively here.

That means it does not take a direct object. So you say:

• Mi aliĝas = I sign up / I join
• Mi aliĝas al la klubo = I join the club

Notice that with the thing being joined, Esperanto often uses al, not a direct object with -n.

So Mi aliĝas la klubon would be wrong.

This is a very common Esperanto pattern.

• aliĝi = to join, to sign oneself up
• aliĝigi = to cause someone to join, to enroll someone, to sign someone up

The suffix -ig- usually means to make or cause.

So:

• Mi aliĝas = I join
• La bibliotekisto aliĝigas min = The librarian signs me up

In your sentence, the wording is slightly different: instead of using aliĝigas, it says the librarian registers my membership.

Because membrecon is the direct object of registras, so it needs the accusative ending -n.

And because adjectives must agree with the nouns they describe, mia also becomes mian.

So:

• membreco = membership
• membrecon = membership as a direct object
• mia membreco = my membership
• mian membrecon = my membership as a direct object

This is a very important Esperanto rule:

• nouns take -n in the accusative
• adjectives modifying those nouns also take -n

• membro = member
• membreco = membership

The suffix -ec- creates an abstract quality, state, or condition.

So:

• bela = beautiful
• beleco = beauty

Similarly:

• membro = a member
• membreco = the state or status of being a member, that is, membership

So registras mian membrecon focuses on registering the membership itself, not the person as an individual.

Ĉe usually means at, by, or with, depending on context.

Here, ĉe la giĉeto means at the counter, at the service window, or at the desk.

This is different from some other prepositions:

• en = in, inside
• al = to, toward
• sur = on
• ĉe = at, by, in the vicinity of, at the place of

So ĉe la giĉeto is natural because the action happens at that service point.

Giĉeto means a service window, counter, ticket window, or similar public-service desk.

In this sentence, it is the place where the librarian handles the registration.

This is the kind of word you might use for:

• a library counter
• a ticket window
• a service desk
• an office window for paperwork

La is the definite article, like English the.

Esperanto uses only one article:

• la = the

There is no separate word for English a or an.

Here, la bibliotekisto and la giĉeto suggest specific, identifiable things in the situation:

• the librarian involved
• the counter being used

That feels natural because in a library setting, the speaker and listener can usually identify which librarian and which counter are meant.

It is built very regularly:

• biblioteko = library
• -ist- = a person professionally or habitually associated with something
• bibliotekisto = librarian

The suffix -ist- is very common in Esperanto. For example:

• arto = art

• artisto = artist

• piano = piano
• pianisto = pianist

So bibliotekisto is exactly the kind of transparent word Esperanto often creates.

A few important pronunciation points:

• ĝ sounds like the English j in job
• ĉ sounds like the English ch in church
• Esperanto stress is almost always on the second-to-last syllable

So:

• aliĝas is pronounced roughly a-LI-jas
• giĉeto is pronounced roughly gi-CHE-to

Also, the g in giĉeto is a hard g, as in go, not like the English j sound. In Esperanto, plain g is always hard.

Yes. Esperanto word order is fairly flexible, as long as the grammar is clear.

The most neutral order here is the one you have:

• la bibliotekisto registras mian membrecon ĉe la giĉeto

But other orders are possible, for example:

• Ĉe la giĉeto la bibliotekisto registras mian membrecon

That version emphasizes the location a bit more.

Because mian membrecon has the accusative ending -n, its role stays clear even if the order changes.

Because registras and registriĝas mean different things.

• registras = registers something
• registriĝas = registers oneself, gets registered

In your sentence, the subject is la bibliotekisto, and the librarian is actively registering mian membrecon. That requires the transitive form registras.

If you said Mi registriĝas, that would mean I register myself or I get registered.

Yes. Esperanto often allows more than one natural expression.

For example, depending on what you want to emphasize, you might also see something like:

• Mi aliĝas; la bibliotekisto registras min kiel membron ĉe la giĉeto.

That means the librarian registers me as a member.

Your original sentence is also perfectly understandable, but it focuses on my membership rather than directly on me. Both patterns are possible, and Esperanto often lets the speaker choose what to spotlight.