Przędziorek jest szkodnikiem.

Because after jest when you say that someone or something is a person, animal, profession, role, or category, Polish very often uses the instrumental case.

• dictionary form: szkodnik
• instrumental singular: szkodnikiem

So:

• Przędziorek jest szkodnikiem. = The spider mite is a pest.

This is one of the most important patterns in Polish:

• X jest Y-em / Y-ą / Y-iem

For example:

• On jest nauczycielem. = He is a teacher.
• To jest problemem. is not natural in the same way, because not every sentence with to be works identically, but with classifying nouns this pattern is very common.

• Przędziorek is in the nominative case, because it is the subject of the sentence.
• szkodnikiem is in the instrumental case, because it is the predicate noun after jest.

So the structure is:

• Przędziorek = nominative subject
• jest = is
• szkodnikiem = instrumental predicate noun

This is a very typical Polish sentence pattern.

Yes. Przędziorek is a masculine singular noun.

You can tell partly from its form:

• it ends in a consonant, which is very often a sign of a masculine noun in Polish

Because it is masculine singular:

• jest is singular
• other words referring back to it would also usually be masculine singular

For example:

• Przędziorek jest mały.
• To groźny szkodnik.

A native English speaker may find the beginning the hardest.

A rough guide:

• prz sounds somewhat like psh or pzh, but not exactly
• ę is a nasal vowel, somewhat like a vowel followed by a light nasal sound
• dzi sounds like a soft jee/dgee sound
• o is like o
• rek ends with a rolled or tapped r and ek

A very rough English-style approximation could be:

pshen-JO-rek or pzhen-JO-rek

But that is only approximate. A few key points:

• rz in modern standard Polish sounds like ż
• dzio in dzio is a soft sound, not a hard English d
• r should be tapped or rolled if possible

A rough approximation is:

shkod-NEE-kyem

Useful parts:

• sz = English sh
• ko = ko
• dni = the dn cluster may feel awkward at first; the ni is soft
• kiem = roughly kyem

The ending -iem is common in masculine instrumental forms.

In standard Polish, jest normally stays in this kind of sentence.

So:

• Przędziorek jest szkodnikiem. = correct

If you remove jest:

• Przędziorek szkodnikiem.

that sounds incomplete in normal standard Polish.

Polish sometimes omits to be in headlines, notes, or very compressed style, but in an ordinary full sentence like this, jest should be there.

The neutral word order is:

• Przędziorek jest szkodnikiem.

But Polish word order is more flexible than English, because case endings show grammatical roles.

You might also see:

• Szkodnikiem jest przędziorek.
• Jest szkodnikiem, ten przędziorek. — more marked, context-dependent

The original order is the most natural if you are simply stating a fact.

Polish has no articles like English a and the.

So:

• Przędziorek jest szkodnikiem.

can mean:

• The spider mite is a pest.
• A spider mite is a pest.

Which one is meant depends on context.

This is normal in Polish. You do not add a separate word for a or the.

Yes, that is also possible, but it is a slightly different structure.

Compare:

• Przędziorek jest szkodnikiem.
• Przędziorek to szkodnik.

Both can mean essentially the same thing here.

The first sentence:

• uses jest
  • instrumental
• is a very standard way to classify something

The second sentence:

• uses to
  • nominative
• is also common, especially in definitions, identifications, and simple statements of equivalence

A learner should know both patterns:

• X jest Y-em
• X to Y

The dictionary form is szkodnik.

That is the nominative singular form, the one you would find in a dictionary.

Here is the change:

• szkodnik → nominative singular
• szkodnikiem → instrumental singular

This kind of change is very common in Polish, so it is worth getting used to seeing nouns in forms other than the dictionary form.

The plural would be:

• Przędziorki są szkodnikami.

Breakdown:

• przędziorek → przędziorki = spider mites
• jest → są = are
• szkodnikiem → szkodnikami = pests, in the instrumental plural

So the same pattern stays the same:

• subject in nominative
• predicate noun in instrumental

That is because of the way certain masculine nouns form the instrumental singular.

For a noun like szkodnik, the instrumental is:

• szkodnikiem

Many masculine nouns behave similarly:

• nauczyciel → nauczycielem
• student → studentem
• robotnik → robotnikiem

So -iem is a normal instrumental ending for many masculine nouns, especially when the stem shape calls for it.

The important thing for a learner is to recognize the whole form:

• szkodnikiem = instrumental of szkodnik

No. In Polish, common nouns are not capitalized.

So:

• przędziorek
• szkodnik

both stay lowercase unless they begin a sentence.

This is different from German, not from English. Proper names still use capitals, of course, but ordinary nouns do not.