Der Spielzug bewegt die Spielfigur vorwärts.

Because the two nouns have different grammatical genders:

• der Spielzug is masculine
• die Spielfigur is feminine

In German, every noun has a grammatical gender, and the article has to match it.

In this sentence:

• der Spielzug = the subject
• die Spielfigur = the object

A useful extra point: die can be both:

• feminine nominative
• feminine accusative

So die Spielfigur looks the same in both cases. You recognize it as the object mainly from the sentence structure and meaning.

The subject is der Spielzug, and the object is die Spielfigur.

A native English speaker often asks this because English relies heavily on word order, while German also uses case.

Here, several clues help:

1. The verb form is bewegt, which is third-person singular.
   That matches der Spielzug = it moves.

2. Der Spielzug is in the nominative case, which is the usual case for the subject.

3. Die Spielfigur is the thing being moved, so it is the direct object in the accusative.

So the structure is:

• Der Spielzug = subject
• bewegt = verb
• die Spielfigur = direct object
• vorwärts = adverb

Bewegt is the third-person singular present tense form of the verb bewegen.

The infinitive is:

• bewegen = to move

The present-tense forms are:

• ich bewege
• du bewegst
• er/sie/es bewegt
• wir bewegen
• ihr bewegt
• sie/Sie bewegen

Since der Spielzug is singular, German uses bewegt.

Because German main clauses normally follow the V2 rule: the finite verb comes in the second position.

In this sentence:

• Der Spielzug = first element
• bewegt = second element

So the pattern is:

Subject + verb + object + adverb

That is very common in German main clauses.

You could also rearrange the sentence for emphasis, but the finite verb still stays second:

• Vorwärts bewegt der Spielzug die Spielfigur.

That sounds more marked or stylistic, but it follows the same rule.

Because vorwärts is an adverb, not a noun phrase.

It directly describes how / in what direction the piece is moved:

• vorwärts = forward / forwards

German often uses a simple adverb where English might use a slightly different structure.

So:

• die Spielfigur vorwärts bewegen = to move the game piece forward

No extra word like to is needed.

Vorwärts is an adverb of direction.

It tells you the direction of the movement:

• vorwärts = forward
• rückwärts = backward

In this sentence, it modifies the verb bewegt.

So the core idea is:

• bewegt = moves
• vorwärts = forward

Together: moves ... forward

Yes, often you can.

For example:

• Der Spielzug bewegt die Spielfigur nach vorne.

That is also understandable and natural in many contexts.

The difference is mainly one of style and nuance:

• vorwärts is a compact adverb meaning forward
• nach vorne literally means to the front / forward

In game instructions or movement descriptions, vorwärts often sounds neat and direct.

Because this sentence uses bewegen as a transitive verb: it moves something else.

Compare:

• Die Spielfigur bewegt sich.
  = The game piece moves / is moving.
  Here the piece moves itself, so German uses sich bewegen.

• Der Spielzug bewegt die Spielfigur.
  = The move moves the game piece.
  Here something acts on the piece, so German uses plain bewegen with a direct object.

So:

• sich bewegen = to move oneself / to move
• etwas bewegen = to move something

Yes, it is a normal German compound noun.

It is made from:

• Spiel = game
• Zug = move, turn, pull, train, etc. depending on context

In this context, Spielzug means something like:

• move
• play
• turn in a game

German uses a lot of compound nouns, so seeing a long word like this is very normal.

Because German usually writes compound nouns as one single word.

So:

• Spiel
  • Figur = Spielfigur

This is extremely common in German. English often writes similar ideas as two words, but German prefers one compound noun.

Other examples:

• Haus
  • Tür = Haustür
• Kinder
  • Buch = Kinderbuch

So Spielfigur is completely standard.

Because in German, all nouns are capitalized.

That is one of the first things English speakers notice. So:

• der Spielzug
• die Spielfigur

Both are nouns, so both begin with capital letters.

By contrast:

• bewegt is a verb, so it is not capitalized
• vorwärts is an adverb here, so it is not capitalized

Grammatically, yes.

It is clear and correct as a sentence meaning that a game move causes a game piece to move forward.

That said, depending on the exact context, German speakers might also phrase the idea differently, for example:

• Mit diesem Spielzug wird die Spielfigur vorwärts bewegt.
• Dieser Spielzug bewegt die Spielfigur vorwärts.

The version you have is still useful and grammatically sound for learning.

The plurals are:

• der Spielzug → die Spielzüge
• die Spielfigur → die Spielfiguren

Examples:

• Die Spielzüge sind wichtig.
  = The moves are important.

• Die Spielfiguren stehen auf dem Brett.
  = The game pieces are on the board.

This is worth learning early, because German articles change in the plural:

• singular masculine: der
• singular feminine: die
• plural: die for both