Záchranná vesta je vedle vodotěsného vaku.

Because the preposition vedle (next to / beside) requires the genitive case in Czech.

So:

• vodotěsný vak = nominative, the basic dictionary form
• vedle vodotěsného vaku = genitive, because it comes after vedle

This is a very common thing in Czech: prepositions often force a specific case.

Vedle takes the genitive.

Examples:

• vedle domu = next to the house
• vedle auta = next to the car
• vedle vodotěsného vaku = next to the waterproof bag

So if you use vedle, the noun after it should be in the genitive.

Because záchranná agrees with vesta, and vesta is a feminine singular noun.

In Czech, adjectives must agree with the noun in:

• gender
• number
• case

Here:

• vesta = feminine singular nominative
• so the adjective is záchranná

Compare:

• záchranná vesta = life jacket
• záchranný člun = lifeboat
• záchranné lano = rescue rope

The ending changes depending on the noun.

Because it agrees with vaku, which is:

• masculine inanimate
• singular
• genitive

So:

• nominative: vodotěsný vak
• genitive: vodotěsného vaku

Again, the adjective changes to match the noun’s gender, number, and case.

Yes.

• vesta is feminine
• vak is masculine inanimate

A useful beginner clue is the ending:

• nouns ending in -a are often feminine: vesta
• nouns ending in a consonant are often masculine: vak

This is not a perfect rule, but it works often enough to be helpful.

In this sentence, je is the present tense of být (to be) and it is normally included.

• Záchranná vesta je vedle vodotěsného vaku. = full sentence
• Záchranná vesta vedle vodotěsného vaku. = sounds incomplete as a normal statement

Czech sometimes omits to be in headlines, labels, or very compressed styles, but in an ordinary sentence like this, je should be there.

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

You can also say:

• Vedle vodotěsného vaku je záchranná vesta.

Both versions are grammatical. The difference is mainly about focus or emphasis:

• Záchranná vesta je vedle vodotěsného vaku.
  Focus starts with the life jacket.
• Vedle vodotěsného vaku je záchranná vesta.
  Focus starts with the location.

So the original sentence is natural, but it is not the only possible order.

Czech does not have articles like English a/an and the.

So vesta can mean:

• a vest
• the vest

And in context, záchranná vesta can mean:

• a life jacket
• the life jacket

Czech usually relies on context, word order, and emphasis instead of articles.

The dictionary forms are:

• záchranná → záchranný
• vesta → vesta
• je → být
• vedle → vedle
• vodotěsného → vodotěsný
• vaku → vak

This is useful because when you look words up, adjectives and nouns may appear in a different form from the one used in the sentence.

Not exactly.

• vak = sack, bag, sometimes a soft bag or stuff sack
• taška = bag, shopping bag, handbag, general bag
• batoh = backpack

So vodotěsný vak suggests a waterproof bag/sack, often the kind used outdoors or on the water.

A rough pronunciation guide for an English speaker:

• záchranná ≈ ZAAKH-ran-naa
• vesta ≈ VES-ta
• je ≈ ye
• vedle ≈ VED-le
• vodotěsného ≈ vo-do-TYES-ne-ho
• vaku ≈ VA-koo

A few helpful notes:

• á and é are long vowels
• ch is one sound, like the ch in Scottish loch or German Bach
• j is pronounced like English y
• stress in Czech is usually on the first syllable of each word

Usually, yes, in sentences like this.

Vedle means next to / beside and usually suggests something fairly close, often immediately adjacent.

Compare:

• vedle = next to, beside
• blízko = near, close to

So vedle vodotěsného vaku is more naturally next to the waterproof bag than just near the waterproof bag.

Not with exactly the same meaning.

• vedle = next to, beside
• u = by, at, near

So:

• je vedle vodotěsného vaku = it is right next to the waterproof bag
• je u vodotěsného vaku = it is by/near the waterproof bag

Both can be correct in some situations, but vedle is more specific about position.