Muszę nakarmić psa dziś wieczorem.

In Polish, the verb usually already shows who is doing the action, so the subject pronoun is often omitted.

• muszę = I must / I have to
• The ending -ę tells you it is 1st person singular: I

So:

• Muszę nakarmić psa. = I have to feed the dog.
• Ja muszę nakarmić psa. also works, but ja adds emphasis, like I have to feed the dog.

Polish often leaves out pronouns unless they are needed for contrast or emphasis.

Muszę means I must or I have to.

It comes from the verb musieć = to have to / must.

A few useful forms:

• muszę = I must
• musisz = you must
• musi = he/she/it must
• musimy = we must
• musicie = you all must
• muszą = they must

So Muszę nakarmić psa dziś wieczorem literally means something like:

• I must feed the dog this evening

This is about aspect, which is very important in Polish.

• karmić = to feed, imperfective
  
  • focuses on the process, repetition, or ongoing action
• nakarmić = to feed completely / to feed successfully, perfective
  
  • focuses on completing the action

In this sentence, nakarmić is used because the speaker means a completed task: the dog needs to get fed.

Compare:

• Muszę karmić psa codziennie.
  = I have to feed the dog every day.
  (habit / repeated action)

• Muszę nakarmić psa dziś wieczorem.
  = I have to feed the dog this evening.
  (one completed action)

So nakarmić fits better here because it refers to a specific obligation to finish the feeding.

Because psa is the accusative case of pies.

• pies = dog (dictionary form, nominative)
• psa = dog as the direct object

In this sentence, the dog is the thing being fed, so Polish uses the accusative:

• nakarmić psa = to feed the dog

This is especially common with animate masculine nouns, where the accusative often looks like the genitive.

Compare:

• Pies śpi. = The dog is sleeping.
  (pies = subject)

• Widzę psa. = I see the dog.
  (psa = object)

• Muszę nakarmić psa. = I have to feed the dog.
  (psa = object)

Wieczorem is the form used to mean in the evening / this evening.

It comes from wieczór = evening, but here Polish uses the instrumental form:

• wieczór = evening
• wieczorem = in the evening

This is a very common time expression in Polish:

• rano = in the morning
• po południu = in the afternoon
• wieczorem = in the evening
• nocą = at night

So:

• dziś wieczorem = this evening / tonight

They both mean today.

• dziś = shorter, very common
• dzisiaj = also very common, slightly fuller in style

In most situations, they are interchangeable:

• Muszę nakarmić psa dziś wieczorem.
• Muszę nakarmić psa dzisiaj wieczorem.

Both are natural.
Dziś may sound a little more compact, but there is no major difference in meaning here.

Yes. Polish word order is more flexible than English because endings show grammatical roles.

The neutral sentence is:

• Muszę nakarmić psa dziś wieczorem.

But you could also say:

• Dziś wieczorem muszę nakarmić psa.
  Emphasizes this evening
• Psa muszę nakarmić dziś wieczorem.
  Emphasizes the dog
• Muszę dziś wieczorem nakarmić psa.
  Also very natural

The meaning stays basically the same, but the focus changes depending on what comes earlier or gets stressed.

It can often match either one, depending on context.

• dziś wieczorem literally = today evening / this evening
• In natural English, that may be translated as this evening or tonight

So if the meaning shown to the learner is I have to feed the dog tonight, that is perfectly fine.

Polish does not always divide this evening and tonight in exactly the same way English does.

A rough pronunciation guide:

• muszę ≈ MOO-sheh
• nakarmić ≈ nah-KAR-meech
• psa ≈ psah
• dziś ≈ something like jeesh (very rough)
• wieczorem ≈ vye-CHO-rem

A few helpful notes:

• sz sounds like sh
• ć is a soft sound, roughly like a softened ch
• dz and dzi can be tricky for English speakers
• w in Polish sounds like English v

Stress in Polish usually falls on the second-to-last syllable:

• mu-szę
• na-kar-mić
• wie-czo-rem

You could, but it would usually sound less natural in this context.

• Muszę nakarmić psa dziś wieczorem
  = I need to make sure the dog gets fed
  (a single completed task)

• Muszę karmić psa dziś wieczorem
  sounds more like focusing on the activity itself, and in many contexts a native speaker would prefer nakarmić

If you mean one concrete thing on your to-do list, nakarmić is the better choice.

It is neutral and completely natural in everyday Polish.

It is not especially formal and not especially casual. You could use it:

• in conversation
• in a text message
• when talking to family
• when noting a task to yourself

If you wanted to sound slightly more conversational, you might move the time expression:

• Dziś wieczorem muszę nakarmić psa.

But the original sentence is already perfectly normal.