Ο σελιδοδείκτης έπεσε από το βιβλίο, αλλά τον βρήκα κάτω από το τραπέζι.

Ο is the masculine singular definite article in the nominative case, so it means the.

So:

• ο = the for a masculine singular noun as the subject
• σελιδοδείκτης = bookmark

In this sentence, Ο σελιδοδείκτης is the subject of έπεσε (fell), so it appears in the nominative:

• Ο σελιδοδείκτης έπεσε = The bookmark fell

A useful detail: nouns ending in -της are often masculine, and σελιδοδείκτης is one of them.

Έπεσε is the aorist (simple past) of πέφτω (to fall), in the 3rd person singular:

• πέφτω = I fall / I am falling
• έπεσε = he/she/it fell

Here it matches ο σελιδοδείκτης, so:

• Ο σελιδοδείκτης έπεσε = The bookmark fell

The έ- at the beginning is the usual past augment, which often appears in past-tense forms in Greek.

Because από normally takes the accusative case in Modern Greek.

So:

• το βιβλίο = the book
• after από, it stays το βιβλίο

For neuter nouns like βιβλίο, the nominative and accusative singular are often the same, so the form does not visibly change.

That is why you see:

• από το βιβλίο = from the book / out of the book

It can be understood in a few closely related ways, depending on context.

In this sentence:

• Ο σελιδοδείκτης έπεσε από το βιβλίο

English might say:

• The bookmark fell from the book
• The bookmark fell out of the book
• sometimes even fell off the book, depending on the situation

Greek uses από very naturally for this kind of source or starting point. So the Greek is normal, even if English might choose different prepositions depending on the image.

Τον is the masculine singular accusative weak pronoun, meaning him or it, depending on what it refers to.

Here it refers to τον σελιδοδείκτη (the bookmark), which is masculine in Greek. So Greek uses the masculine object pronoun:

• τον βρήκα = I found it

Even though English uses it for a bookmark, Greek matches the grammatical gender of the noun:

• σελιδοδείκτης = masculine
• so the object pronoun is τον

Because weak object pronouns in Modern Greek usually come before the finite verb.

So:

• τον βρήκα = I found him/it
• literally, something like him/it I found

This is standard Greek word order with clitic pronouns. You do not normally say βρήκα τον with the same meaning.

A very common pattern is:

• τον είδα = I saw him/it
• την πήρα = I took her/it
• το βρήκα = I found it

Because Greek often leaves out subject pronouns when they are not needed. The verb ending already tells you who the subject is.

• βρήκα = I found
• βρήκες = you found
• βρήκε = he/she/it found

So Greek does not need to say εγώ unless there is emphasis or contrast.

Here:

• αλλά τον βρήκα κάτω από το τραπέζι = but I found it under the table

If you said αλλά εγώ τον βρήκα..., that would sound more like but I found it..., with extra emphasis on I.

Βρήκα is the aorist of βρίσκω (to find), in the 1st person singular:

• βρίσκω = I find / I am finding
• βρήκα = I found

So:

• τον βρήκα = I found it

This is another simple past form, just like έπεσε.

Yes, κάτω means down, below, or underneath, but very often Greek uses the fixed combination κάτω από to mean under / beneath.

So:

• κάτω = down / below
• κάτω από = under / underneath / beneath

That makes:

• κάτω από το τραπέζι = under the table

This is one of the most common location expressions in Greek.

Compare:

• πάνω στο τραπέζι = on the table
• κάτω από το τραπέζι = under the table

It is in the accusative, because it follows the preposition από in the expression κάτω από.

So:

• κάτω από το τραπέζι = under the table

Again, τραπέζι is a neuter noun, and in the singular its nominative and accusative forms are the same:

• nominative: το τραπέζι
• accusative: το τραπέζι

So the form looks unchanged, even though grammatically it is accusative.

Yes, Greek word order is more flexible than English, but not everything is equally natural.

The version you have is very natural:

• Ο σελιδοδείκτης έπεσε από το βιβλίο, αλλά τον βρήκα κάτω από το τραπέζι.

You could also move some parts for emphasis, for example:

• Κάτω από το τραπέζι τον βρήκα.
  = I found it under the table.
  This gives extra emphasis to under the table.

But the clitic pronoun τον still normally stays before the verb βρήκα.

So Greek allows flexibility, but there are still patterns you need to follow.

Because αλλά means but, and it connects two clauses:

• Ο σελιδοδείκτης έπεσε από το βιβλίο
• αλλά τον βρήκα κάτω από το τραπέζι

Greek normally uses a comma before αλλά in this kind of sentence, just as English often does before but.

Yes, it is a normal word for bookmark.

It is built from:

• σελίδα = page
• δείκτης = pointer / indicator / marker

So σελιδοδείκτης is literally something like a page-marker.

This kind of compound word is very common in Greek.

Because Greek nouns have grammatical gender, and pronouns agree with the noun’s gender, not with natural sex.

So even though a bookmark is an object, Greek treats σελιδοδείκτης as masculine:

• ο σελιδοδείκτης
• τον βρήκα

In English, objects usually become it, but in Greek they can be:

• τον for masculine nouns
• τη(ν) for feminine nouns
• το for neuter nouns

So this is very normal Greek agreement.