Αυτό είναι το επιτραπέζιο παιχνίδι.

Αυτό is the neuter singular form of the demonstrative this.

Greek demonstratives change for gender:

• αυτός = masculine
• αυτή = feminine
• αυτό = neuter

Here, the thing being identified is το επιτραπέζιο παιχνίδι, and παιχνίδι is a neuter noun, so Greek uses αυτό.

A useful pattern is:

• Αυτό είναι το βιβλίο. = This is the book.
• Αυτή είναι η καρέκλα. = This is the chair.
• Αυτός είναι ο φίλος μου. = This is my friend.

Είναι is the present-tense form of είμαι (to be) used for he/she/it and also for this/that.

In this sentence, αυτό takes the same verb form as it, so you say:

• Αυτό είναι ... = This is ...

A few forms of είμαι are:

• είμαι = I am
• είσαι = you are
• είναι = he/she/it is

Greek often omits subject pronouns, but here αυτό is kept because it means this, not just it.

Το is the definite article meaning the.

Since παιχνίδι is a neuter singular noun, its article is το:

• το παιχνίδι = the game

When a noun has an adjective before it, Greek normally puts the article before the adjective:

• το επιτραπέζιο παιχνίδι = the board game

So the structure is:

• το = the
• επιτραπέζιο = board / tabletop
• παιχνίδι = game

This is very normal Greek word order.

Because it is an adjective agreeing with a neuter singular noun.

The dictionary form of the adjective is usually something like:

• επιτραπέζιος, επιτραπέζια, επιτραπέζιο

These are the masculine, feminine, and neuter forms.

Since παιχνίδι is neuter singular, the adjective must also be neuter singular:

• το επιτραπέζιο παιχνίδι

This agreement is a major feature of Greek: adjectives must match the noun in gender, number, and case.

In Greek, every noun has a grammatical gender: masculine, feminine, or neuter.
Παιχνίδι happens to be neuter.

This is usually something you need to learn along with the noun:

• το παιχνίδι = the game

The gender is grammatical, not logical. A thing does not have to be “neutral” in any real-world sense. It is simply how the noun behaves in Greek grammar.

Because παιχνίδι is neuter, other words around it also become neuter:

• το παιχνίδι
• επιτραπέζιο παιχνίδι
• αυτό είναι το επιτραπέζιο παιχνίδι

It is in the nominative case.

After the verb είμαι (to be), Greek uses the nominative for the thing being identified:

• Αυτό είναι το επιτραπέζιο παιχνίδι.

So both:

• Αυτό
• το επιτραπέζιο παιχνίδι

are nominative here.

This is similar to sentences like:

• Αυτό είναι το σπίτι.
• Αυτή είναι η δασκάλα.

Yes, Greek word order is more flexible than English, but the given sentence is the most neutral and natural way to say it.

Normal:

• Αυτό είναι το επιτραπέζιο παιχνίδι.

Other possible orders can change emphasis:

• Το επιτραπέζιο παιχνίδι είναι αυτό.
  = The board game is this one. / This is the one that is the board game.

So although Greek allows movement, not every order sounds equally neutral. For a learner, the original sentence is the best default pattern.

You would usually say:

• Αυτό είναι ένα επιτραπέζιο παιχνίδι.

Here:

• ένα = a / an (neuter singular)

Compare:

• Αυτό είναι το επιτραπέζιο παιχνίδι. = This is the board game.
• Αυτό είναι ένα επιτραπέζιο παιχνίδι. = This is a board game.

So the main change is:

• το → ένα

A rough pronunciation is:

AF-to EE-ne to e-pi-tra-PE-zio peh-KHNI-thee

A few helpful notes:

• αι is pronounced like e in met
• ει is pronounced like ee
• γ before ι sounds like a soft sound, often approximated in learner materials as y or a gentle voiced sound
• χ in παιχνίδι is a throaty sound, not like English k

The stress falls on:

• αυτό
• είναι
• επιτραπέζιο
• παιχνίδι

More or less, yes.

The adjective επιτραπέζιος / επιτραπέζια / επιτραπέζιο comes from the idea of something on the table or tabletop. So επιτραπέζιο παιχνίδι literally suggests a tabletop game, which is the normal Greek expression for board game.

So even if the literal parts do not match English exactly, the phrase is the natural Greek way to express that idea.