Stojánek je vedle blatníku.

Because vedle requires the genitive case.

• Dictionary form: blatník
• After vedle: vedle blatníku

So the sentence uses blatníku because it means next to the fender/mudguard, and vedle changes the following noun into the genitive.

Stojánek is in the nominative singular because it is the subject of the sentence.

In other words, it is the thing that is somewhere:

• Stojánek = the kickstand
• je vedle blatníku = is next to the fender

So the basic structure is:

subject + je + place

Here, vedle means next to, beside, or by.

So:

• vedle blatníku = next to the fender / beside the fender

It is a very common Czech preposition for position in space.

Yes, in normal modern Czech, vedle is followed by the genitive.

Examples:

• vedle domu = next to the house
• vedle auta = next to the car
• vedle blatníku = next to the fender

So this sentence is a good example of a preposition + genitive pattern that you should memorize.

Je means is.

It is the 3rd person singular form of the verb být (to be).

Some forms of být:

• jsem = I am
• jsi = you are
• je = he/she/it is
• jsme = we are
• jste = you are
• jsou = they are

So:

• Stojánek je vedle blatníku.
• The kickstand is next to the fender.

Because Czech does not use articles like English a and the.

So a Czech sentence often leaves that information to context.

• stojánek can mean a kickstand or the kickstand
• blatníku can mean of a fender or of the fender, depending on context

English must choose an article, but Czech usually does not.

Yes, it can be understood as a diminutive or a smaller/specialized form related to stojan (stand).

In practice, stojánek is commonly used for a kickstand, especially on a bicycle. So even if it looks diminutive in form, it often functions as the normal word for that object.

For a learner, the most useful thing is simply to remember:

• stojánek = kickstand

Both are masculine inanimate nouns.

You can tell from their dictionary forms ending in a consonant:

• stojánek
• blatník

That matters because masculine nouns follow certain declension patterns. For example, the genitive singular of blatník is blatníku.

Yes. Czech word order is more flexible than English.

You could also say:

Vedle blatníku je stojánek.

This still means the same thing: The kickstand is next to the fender.

The difference is mainly focus or emphasis:

• Stojánek je vedle blatníku. → focuses first on the kickstand
• Vedle blatníku je stojánek. → focuses first on the location

Both are grammatically correct.

A rough English-friendly guide is:

STOY-aa-nek yeh VED-leh BLAT-nee-koo

A few helpful points:

• Czech stress is usually on the first syllable of each word.
• j is pronounced like English y.
• á is a long a sound.
• í is a long ee sound.

So:

• Stojánek ≈ STOY-aa-nek
• je ≈ yeh
• vedle ≈ VED-leh
• blatníku ≈ BLAT-nee-koo

Yes. Besides the physical meaning next to/beside, vedle can also have broader meanings like besides or in addition to, depending on context.

But in this sentence, the meaning is clearly spatial:

• Stojánek je vedle blatníku.
• The kickstand is next to the fender.

So here you should understand vedle as a location word.