Jeśli zapomnę o pielusze, będę musiała wejść do sklepu po drodze.

Jeśli means if.

So Jeśli zapomnę o pielusze... = If I forget the diaper...

It introduces a condition: first this may happen, and then something else will happen as a result.

Polish also has other words for if/when, but jeśli is the normal straightforward choice here.

Zapomnę is the 1st person singular future of the perfective verb zapomnieć.

In Polish, perfective verbs do not have a true present tense. Their present-looking forms actually refer to the future. So:

• zapomnę = I will forget
• not I forget in the present

That is why after jeśli you can get a form that translates as future in English:

• Jeśli zapomnę... = If I forget...

This is normal in Polish. English usually avoids will after if, but Polish does not work the same way here.

Because Polish often leaves out subject pronouns when the verb already shows who is doing the action.

• zapomnę already means I will forget
• będę musiała already means I will have to

So ja is not necessary.

If you added ja, it would usually be for emphasis or contrast, for example:

• Jeśli ja zapomnę... = If I’m the one who forgets...

After the preposition o, Polish normally uses the locative case.

So:

• basic form: pielusza or more commonly pieluszka depending on the word chosen
• after o: o pielusze

So grammatically:

• o + locative

That is why you get pielusze, not the basic dictionary form.

Yes, literally zapomnieć o czymś means to forget about something.

So zapomnę o pielusze is understandable, but depending on context, many speakers might say something more specific if they mean forget to bring/take the diaper, for example:

• Jeśli zapomnę pieluszki...
• Jeśli zapomnę wziąć pieluszkę...

Those can sound more natural if the idea is I forget to bring it with me.

So the sentence you have is understandable, but zapomnieć o + locative can sound a bit more like forget about than leave behind.

Because musieć is an imperfective verb, and imperfective verbs usually make the future with:

• będę / będziesz / będzie / ...
  plus
• a past-like form or infinitive

Here the pattern is:

• będę musiała = I will have to

This structure is very common in Polish:

• będę robić = I will be doing / I will do
• będę musiała = I will have to

Because the speaker is feminine.

In Polish, forms like musiał / musiała agree with gender:

• musiał = masculine speaker
• musiała = feminine speaker

So:

• będę musiała = I will have to said by a woman
• będę musiał = I will have to said by a man

English does not show this difference, but Polish does.

Wejść is a perfective verb meaning to go in / enter.

So:

• wejść do sklepu = to go into the store / enter the store

It fits well here because the action is seen as a single completed event: going into the shop one time.

A more general or repeated-action version would use an imperfective verb such as wchodzić, but here wejść is the natural choice.

Because do sklepu expresses movement into the store.

Compare:

• wejść do sklepu = to go into the store
• być w sklepie = to be in the store

So:

• do + genitive is often used for movement toward/into a place
• w + locative is used for being inside a place

Here the sentence is about entering the store, so do sklepu is correct.

Po drodze means on the way or along the way.

In this sentence:

• wejść do sklepu po drodze = go into a store on the way

It is a very common expression. It often means you stop somewhere while going somewhere else.

Examples:

• Wpadnę po drodze. = I’ll stop by on the way.
• Kupimy to po drodze. = We’ll buy it on the way.

So the full idea is that if the speaker forgets the diaper, she will have to stop at a store while already on the way somewhere.