Υπάρχει ένα ψίχουλο στο τραπέζι.

Yes. Υπάρχει is the 3rd person singular form of υπάρχω, and in sentences like this it means there is / exists.

Greek does not use a separate dummy subject like English there. So instead of saying something like there is, Greek simply says exists:

• Υπάρχει ένα ψίχουλο... = There is a crumb...

If the noun were plural, you would use υπάρχουν:

• Υπάρχουν ψίχουλα στο τραπέζι. = There are crumbs on the table.

Ένα here is the neuter singular indefinite article, so it usually means a/an.

It can also mean one, depending on context, but in this sentence it is most naturally just a.

Why ένα specifically?

• ψίχουλο is a neuter noun
• the indefinite article must match the noun’s gender and number

So:

• masculine: ένας
• feminine: μία / μια
• neuter: ένα

Because ψίχουλο is neuter, you get ένα ψίχουλο.

Ψίχουλο is neuter.

A helpful clue is the ending -ο, which is very often neuter in Greek. That is why the article is ένα.

So:

• το ψίχουλο = the crumb
• ένα ψίχουλο = a crumb

As always, article and noun agree in gender, number, and case.

The word order υπάρχει ένα ψίχουλο στο τραπέζι is the most neutral and natural way to present new information: there is a crumb on the table.

Greek word order is more flexible than English, so other orders are possible, but they change the emphasis:

• Υπάρχει ένα ψίχουλο στο τραπέζι.
  Neutral: there is a crumb on the table.

• Στο τραπέζι υπάρχει ένα ψίχουλο.
  Emphasis on on the table.

• Ένα ψίχουλο υπάρχει στο τραπέζι.
  More marked; emphasis on a crumb.

So the given order is the standard, unmarked one.

Στο means on the / in the / at the, depending on context.

It is a contraction of:

• σε = in, at, to, on
• το = the (neuter singular)

So:

• σε + το = στο

In this sentence:

• στο τραπέζι = on the table

This kind of contraction is very common in Greek:

• στον = σε + τον
• στην = σε + την
• στα = σε + τα

Greek prepositions like σε normally take the accusative case.

So in στο τραπέζι, the noun is in the accusative singular. The reason it looks unchanged is that many neuter nouns have the same form in the nominative and accusative singular.

For τραπέζι:

• nominative: το τραπέζι
• accusative: το τραπέζι

So the form does not visibly change, even though the case is accusative.

This is an important difference.

Use υπάρχει when you mean that something exists / is present somewhere:

• Υπάρχει ένα ψίχουλο στο τραπέζι.
  = There is a crumb on the table.

Use είναι when you are talking about where a specific, known thing is:

• Το ψίχουλο είναι στο τραπέζι.
  = The crumb is on the table.

So:

• υπάρχει introduces something into the scene
• είναι describes or locates something already identified

You can, but it changes the feel.

• Υπάρχει ένα ψίχουλο στο τραπέζι.
  = There is a crumb on the table.
  More straightforward and natural in many contexts.

• Υπάρχει ψίχουλο στο τραπέζι.
  = There is crumb / there are some crumbs/crumb matter on the table.
  This can sound less specific or more like talking about the substance in general.

In many everyday situations, the version with ένα is what a learner should use first.

A simple pronunciation guide:

• Υπάρχει → ee-PAR-khee
• ένα → E-na
• ψίχουλο → PSEE-khou-lo
• στο → sto
• τραπέζι → tra-PE-zi

A few notes:

• ψ sounds like ps in lips
• χ is a rough sound, like German Bach or Scottish loch
• the stressed syllable is marked by the accent:
  • υπάρχει
  • ένα
  • ψίχουλο
  • τραπέζι

You would change both the verb and the noun phrase:

• Υπάρχουν ψίχουλα στο τραπέζι.
  = There are crumbs on the table.

Changes:

• υπάρχει → υπάρχουν because the noun is now plural
• ψίχουλο → ψίχουλα (plural)

You could also say:

• Υπάρχουν μερικά ψίχουλα στο τραπέζι.
  = There are some crumbs on the table.

For a question, Greek usually keeps the same word order and uses intonation, or a question mark in writing:

• Υπάρχει ένα ψίχουλο στο τραπέζι;
  = Is there a crumb on the table?

For a negative, use δεν before the verb:

• Δεν υπάρχει ένα ψίχουλο στο τραπέζι.
  = There isn’t a crumb on the table.

Very often, Greek would more naturally say:

• Δεν υπάρχει ψίχουλο στο τραπέζι.
  = There isn’t any crumb on the table / There are no crumbs on the table.

That version sounds more idiomatic in many situations.