Tato stavba je stará.

Because stavba is a feminine singular noun in the nominative case.

The demonstrative this changes form in Czech to match the noun:

• masculine: tento
• feminine: tato
• neuter: toto

So:

• tato stavba = this structure/building

You may also see the shorter everyday form ta stavba, which often means that building or sometimes just the building / this building depending on context. But in careful standard Czech, tato stavba is the clear form for this building/structure.

Because adjectives in Czech must agree with the noun they describe.

Since stavba is:

• feminine
• singular
• nominative

the adjective starý (old) must take the matching feminine nominative singular form:

• masculine: starý
• feminine: stará
• neuter: staré

So:

• stará stavba = an old structure/building
• Tato stavba je stará. = This structure/building is old.

It is in the nominative case.

That is because stavba is the subject of the sentence:

• Tato stavba = the thing being talked about
• je stará = is old

Both tato and stará also appear in forms that match the nominative feminine singular noun stavba.

Not exactly.

Stavba is a broader word. It can mean:

• a structure
• a building
• a construction
• sometimes even the act/process of construction, depending on context

Budova usually means a building more specifically, especially as a physical building people use.

So stavba can feel a bit broader or more technical than budova. In this sentence, it could be understood as building or structure, depending on context.

Je is the 3rd person singular form of být (to be), meaning is.

So:

• je = is

In standard Czech, you normally keep it in a sentence like this:

• Tato stavba je stará.

Unlike in Russian, for example, the present-tense to be is not normally left out in standard Czech.

In casual speech, people may shorten or reduce things, but for learners the correct model is definitely to keep je here.

No, Czech word order is fairly flexible, but different orders change the focus or emphasis.

The neutral order is:

• Tato stavba je stará.

Other orders are possible, for example:

• Stará je tato stavba.
• Je tato stavba stará? = Is this building old?

Even when the grammar stays correct, changing the order can sound marked, contrastive, or question-like. For learners, the basic order subject + je + adjective is the safest choice.

A simple pronunciation guide is:

• Tato ≈ TAH-to
• stavba ≈ STAV-ba
• je ≈ ye
• stará ≈ STAH-raa

A few helpful points:

• Czech stress is usually on the first syllable of each word.
• j is pronounced like English y in yes.
• á is a long vowel, so it is held a bit longer than short a.

So stará is not the same as stara; the long á matters.

Czech has no articles like English a/an and the.

That means Czech often just says the noun directly:

• stavba can mean a building, the building, or just building/structure, depending on context

In this sentence, the demonstrative tato already makes it specific:

• tato stavba = this building/structure

So there is no need for an article.

Yes, it could, depending on context.

Because stavba is broader than just building, the sentence might refer to:

• a building
• a structure
• a construction project
• a built object in general

If the context is architecture, engineering, or historical monuments, structure may be a better translation than building.

Yes. In everyday speech, people often use the shorter demonstrative:

• Ta stavba je stará.

This is very natural in conversation.

The difference is roughly:

• tato stavba = more explicit, standard, slightly more formal or careful
• ta stavba = very common in everyday spoken Czech

For a learner, both are useful, but tato stavba je stará is a very clear textbook-style example of agreement.

Because of the verb je.

Compare:

• stará stavba = an old structure
  Here stará directly modifies the noun.

• stavba je stará = the structure is old
  Here stará comes after je and describes the subject through the verb to be.

In both cases, the adjective still agrees with stavba, but the sentence structure is different:

• attributive adjective: stará stavba
• predicate adjective: stavba je stará

In your sentence, it is clearly the second type.