Эта семья бедная.

Because эта must agree with семья in gender, number, and case.

• семья is a feminine singular noun
• so the demonstrative this must also be feminine singular
• that gives эта семья = this family

Compare:

• этот = masculine singular
• эта = feminine singular
• это = neuter singular
• эти = plural

In Russian, every noun has a grammatical gender. Семья is grammatically feminine, even though a family includes people of different sexes.

A common clue is the ending: many nouns ending in -я or -а are feminine. So семья is treated like a feminine singular noun, and any adjectives or demonstratives that go with it must match that.

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

Here, бедная describes семья, so it also has to be:

• feminine
• singular
• nominative

That is why you get бедная.

The basic dictionary form is бедный = poor, but it changes:

• бедный = masculine
• бедная = feminine
• бедное = neuter
• бедные = plural

So Эта семья бедная uses the feminine form because семья is feminine.

In the present tense, Russian usually does not use a verb equivalent to is/am/are.

So:

• Эта семья бедная literally looks like This family poor
• but it naturally means This family is poor

This is completely normal in Russian.

In other tenses, Russian does use a form of to be:

• Эта семья была бедной = This family was poor
• Эта семья будет бедной = This family will be poor

The sentence uses the nominative case.

• эта is nominative feminine singular
• семья is nominative singular
• бедная is nominative feminine singular

This is because семья is the subject, and бедная is a predicate adjective agreeing with it.

Yes. Russian word order is more flexible than English word order.

The neutral, ordinary order is:

Эта семья бедная.

But other orders are possible for emphasis or style, for example:

• Бедная эта семья. = emphasis on poor
• Семья эта бедная. = more marked or contrastive

Even though word order can change, the most natural basic version for learners is Эта семья бедная.

Yes, but the style is different.

• бедная is the full form adjective
• бедна is the short form

Both can mean poor, but they are not always identical in tone.

For this kind of simple statement, Эта семья бедная sounds very normal and straightforward.
Эта семья бедна is also correct, but it can sound a bit more formal, literary, or concise.

For many learners, it is safest to start with the full form: бедная.

Because бедно is not the form that agrees with семья.

Бедно is usually:

• the neuter singular adjective form, or
• an adverb meaning something like poorly or in a poor way, depending on context

But семья is feminine, so the adjective must be feminine too:

• семья бедная not
• семья бедно

Семья is pronounced approximately sim-YA.

A few helpful points:

• the stress is on the last syllable: семЬЯ
• the ь is a soft sign; it does not have its own sound, but it softens the consonant before it
• the я after ь is pronounced like ya

So it is roughly:

• сем
  • ья
• sim-YA

It is pronounced roughly BYED-na-ya.

Helpful points:

• the stress is on the first syllable: бЕдная
• е after б sounds like ye
• the ending -ая is pronounced like a-ya, though in fast speech it can sound a bit reduced

A learner-friendly approximation is BYED-na-ya.

It specifically means this family.

• эта = this for a feminine singular noun
• so эта семья points to a particular family near the speaker, already known in context, or being singled out

If you just want to say the family in a general way, Russian often simply uses семья, because Russian has no articles a/an/the.

So:

• семья can mean a family or the family, depending on context
• эта семья clearly means this family

Usually in this sentence, yes: бедная most naturally means poor in the financial sense.

But in other contexts, бедный / бедная can also mean poor in the emotional sense, like poor thing or unfortunate.

For example:

• Бедная девочка! = Poor girl!

In Эта семья бедная, the most natural meaning is that the family does not have much money.