In foro figulus vasa vendit et puellae monstrat quomodo ex luto vas fingat.

Latin has no articles. There is no separate word for the or a/an.

So:

• figulus can mean the potter or a potter
• vas can mean the pot or a pot
• puellae here can mean to the girl or to a girl

Context tells you which is most natural.

Because in with the meaning in/on/at (location) takes the ablative.

• forum = nominative/accusative singular
• foro = ablative singular

So in foro means in the forum / in the marketplace.

If there were motion into the forum, Latin would normally use in forum with the accusative.

Figulus is nominative singular, which is the normal case for the subject of a clause.

It matches the verbs:

• vendit = he/she sells
• monstrat = he/she shows

So figulus is the person doing both actions.

Because vasa is a neuter plural form.

The noun is:

• singular: vas
• plural: vasa

A very important Latin pattern is:

• neuter nominative plural often ends in -a
• neuter accusative plural also often ends in -a

Here vasa is the accusative plural, the direct object of vendit.

Here puellae is best understood as dative singular: to the girl.

That is because monstrat often takes:

• the person shown to in the dative
• the thing shown as a noun or a clause

So:

• puellae monstrat = he shows to the girl

This form puellae could look ambiguous by itself, since it can also be:

• nominative plural
• genitive singular
• dative singular

But in this sentence, the grammar points to dative singular.

Because Latin often uses the dative case by itself for an indirect object.

English usually says:

• he shows it to the girl

Latin can simply say:

• puellae monstrat

So the ending itself tells you to the girl.

The thing being shown is the whole clause:

quomodo ex luto vas fingat

In other words, Latin can use an entire clause as the object of a verb like show, know, ask, or say.

So the structure is roughly:

• he shows the girl [how he shapes a pot from clay]

Quomodo means how.

It introduces an indirect question:

• direct: Quomodo vas fingit? = How does he shape a pot?
• indirect: monstrat quomodo vas fingat = he shows how he shapes a pot

So it is not a separate statement; it is a dependent clause after monstrat.

Because quomodo ... fingat is an indirect question, and Latin normally uses the subjunctive in indirect questions.

So:

• fingit = indicative
• fingat = subjunctive

Here fingat is present subjunctive.

That does not mean “may shape” in this sentence. It is subjunctive because of the grammar of the indirect question, not because the action is uncertain.

It is present subjunctive because the main verb is also present:

• monstrat

In classical Latin sequence of tenses, after a primary tense such as monstrat, a contemporaneous action in an indirect question is usually shown by the present subjunctive.

So:

• monstrat ... fingat = he shows how he is shaping / shapes

If the main verb were past, you would often expect an imperfect subjunctive instead:

• monstrabat ... fingeret

The subject is understood from the context, and it is most naturally the same potter.

Latin often leaves subject pronouns unstated because the verb ending already shows the person:

• fingat = he/she may shape in form, but here simply he shapes within the indirect question

So the sense is:

• the potter sells pots
• and shows the girl how he shapes a pot from clay

Ex means out of / from, and it takes the ablative.

So:

• lutum = clay, mud
• luto = ablative singular

Therefore:

• ex luto = out of clay / from clay

Because the sentence is talking about two different numbers:

• vasa vendit = he sells pots in general, so plural
• vas fingat = he shows how he shapes a pot, so singular

That is perfectly normal. A person may sell many pots, but demonstrate the making of just one pot.

Latin word order is much more flexible than English word order, because the endings show the grammatical roles.

This sentence is arranged in a natural but not rigid way:

• In foro sets the scene
• figulus gives the subject
• vasa vendit gives the first action
• et puellae monstrat gives the second action
• quomodo ex luto vas fingat comes last as the clause being explained

A different order could still make sense, though the emphasis might change.

Yes, they are related in meaning.

• figulus = potter
• fingere = to shape, form, mold
• fingat = a form of fingere

So there is a nice word-family connection: the figulus is literally the person who shapes things, especially clay vessels.