Το επιτραπέζιο παιχνίδι έχει ταμπλό, πιόνι και ζάρι.

Το is the definite article meaning the. It agrees with παιχνίδι because παιχνίδι is neuter singular.

So:

• το = the for a neuter singular noun
• παιχνίδι = game
• το παιχνίδι = the game

Because επιτραπέζιο παιχνίδι is one neuter singular thing, the article is το.

Επιτραπέζιο means tabletop or board-game-related here.

Literally, it is connected to τραπέζι meaning table. So επιτραπέζιο παιχνίδι is basically a game played on a table, which is why it means board game.

You will often see:

• παιχνίδι = game
• επιτραπέζιο παιχνίδι = board game

In Greek, when an adjective comes before a noun, it is very normal to use just one article before the adjective:

• το επιτραπέζιο παιχνίδι

This is a standard pattern:
article + adjective + noun

Greek can also sometimes use another pattern with a repeated article, but that is not needed here. For a learner, the important thing is that το επιτραπέζιο παιχνίδι is completely normal and natural.

Because the subject is singular:

• Το επιτραπέζιο παιχνίδι = The board game

So the verb must also be singular:

• έχει = it has

If the subject were plural, then you would use έχουν:

• Τα επιτραπέζια παιχνίδια έχουν... = Board games have...

They are singular because the sentence is listing the basic components of the game in a general way:

• a board
• a piece/token
• a die

Greek often does this when describing what something comes with or what it consists of.

If you wanted to emphasize that there are several pieces or several dice, you could use plurals instead:

• πιόνια = pieces/tokens
• ζάρια = dice

So the singular here is not strange; it is just a simple, general description.

Greek often leaves out the indefinite article in lists like this, especially when naming parts, contents, or typical features.

So:

• έχει ταμπλό, πιόνι και ζάρι

sounds natural as it has a board, a piece, and a die

You could sometimes add ένα for emphasis or specificity, but it is not necessary here. Omitting it sounds neat and idiomatic.

Not only that. Πιόνι can mean a pawn in chess, but it can also mean a playing piece, token, or counter in other board games.

So in this sentence, πιόνι should be understood more generally as a game piece, not necessarily specifically a chess pawn.

Yes. They are the direct objects of έχει, so they are in the accusative.

However, these are neuter singular nouns, and in Greek, neuter singular nouns often have the same form in the nominative and accusative.

So:

• το ταμπλό can be nominative or accusative
• το πιόνι can be nominative or accusative
• το ζάρι can be nominative or accusative

That is why their form does not visibly change here.

Και means and.

The list works just like in English:

• ταμπλό, πιόνι και ζάρι
• board, piece and die

Greek normally uses commas between earlier items and και before the last one, just as English does.

A simple learner-friendly pronunciation is:

To epi-tra-PE-zio peh-HNI-thi E-hi tam-BLO, pee-O-ni ke ZA-ri.

A few helpful notes:

• χ in έχει is a rough sound, not like English h exactly.
• αι in και sounds like e in bed
• The stress marks matter:
  • επιτραπέζιο
  • παιχνίδι
  • ταμπλό
  • πιόνι
  • ζάρι

If you want to sound more natural, pay special attention to where the stress falls in each word.