Learn Polish: Nasza walizka jest bardzo ciężka.

In Polish, words like 'our' have to match the gender of the noun they belong to. Because walizka (suitcase) is a feminine noun (notice it ends in -a), we use the feminine form nasza. If it were a masculine noun like pies (dog), you would use nasz.

Why does the adjective ciężka (heavy) also end in -a?

Adjectives also change their endings to match the noun's gender. Since walizka is feminine, the adjective takes the feminine ending -a. If you looked up 'heavy' in a dictionary, you would find the masculine form, ciężki.

Does the word bardzo (very) need to match the feminine noun too?

No! Bardzo is an adverb, which means it describes how heavy the suitcase is, not the suitcase itself. Adverbs in Polish never change their shape, so bardzo stays exactly the same no matter what noun you use.

Is the word order fixed, or could I say Walizka nasza jest bardzo ciężka?

Just like in English, possessive words like nasza almost always go right before the noun they describe. Putting nasza after the noun is understood, but it sounds unnatural or overly poetic in everyday conversation.