El artista firma el cuadro.

Because artista is a noun of common gender. That means the word form stays the same for a male or female person, and the article tells you the gender:

• el artista = the male artist
• la artista = the female artist

So the -a ending does not always mean a noun is feminine in Spanish.

Spanish uses the definite article el with both nouns here because both are specific:

• el artista = the artist
• el cuadro = the painting / picture

In English, we also often say The artist signs the painting. Spanish simply keeps the article with each noun where it is needed.

Spanish often leaves out subject pronouns because the verb ending already shows who is doing the action.

Here:

• firma = he/she signs

So El artista firma already clearly means The artist signs. Adding él would usually be unnecessary unless you want extra emphasis or contrast.

Firma is the third person singular present tense of firmar.

Conjugation of firmar in the present:

• yo firmo = I sign
• tú firmas = you sign
• él/ella/usted firma = he/she/you formal signs
• nosotros firmamos = we sign
• vosotros firmáis = you all sign
• ellos/ellas/ustedes firman = they/you all sign

So El artista firma el cuadro means The artist signs the painting or sometimes is signing, depending on context.

In this sentence, firma is a verb: signs.

But firma can also be a noun:

• la firma = the signature
• la firma = the company / firm, in some contexts

So context matters. In El artista firma el cuadro, it is clearly the verb because it comes after the subject el artista and before the object el cuadro.

Here, cuadro means painting or picture, especially a framed work of art.

Depending on context, cuadro can also mean other things, such as:

• a picture
• a painting
• a frame
• a chart / table
• a square

In this sentence, because we have el artista and firma, the most natural meaning is painting or picture.

This is the most neutral and common Spanish word order:

• subject + verb + object

So:

• El artista = subject
• firma = verb
• el cuadro = direct object

Spanish word order is more flexible than English, so other orders are possible for emphasis, style, or in questions. But for a basic statement, El artista firma el cuadro is the standard choice.

The direct object is the thing directly affected by the action of the verb.

Ask:

• What does the artist sign?

Answer:

• el cuadro

So el cuadro is the direct object.

The personal a is normally used before a specific person (and sometimes pets or personified beings) when that person is the direct object.

Examples:

• Veo al artista. = I see the artist.
• Conozco a María. = I know María.

But el cuadro is a thing, not a person, so there is no personal a:

• El artista firma el cuadro.

In Spain, cuadro is pronounced roughly like KWA-dro.

A few useful points:

• cua- sounds like kwa
• the u in qu is silent in spelling combinations like que and qui, but here it is part of cua, so it is pronounced
• dr is pronounced together smoothly
• the stress is on CUA: CUA-dro

Yes. The Spanish present tense often covers both:

• The artist signs the painting
• The artist is signing the painting

The exact English translation depends on context. If Spanish wants to strongly emphasize an action in progress, it can also use:

• El artista está firmando el cuadro. = The artist is signing the painting.

But firma by itself can already express that idea in the right context.