Mein Bruder sucht das Bügeleisen, weil er sein Hemd noch bügeln will.

In German, mein already works like a determiner, so you normally do not add another article before it.

• mein Bruder = my brother
• not der mein Bruder

This is similar to English: we say my brother, not the my brother.

Bügeleisen is a neuter noun in German, so its article is das.

• das Bügeleisen = the iron

Here it is in the accusative case because it is the thing being looked for, but for neuter nouns, das stays das in both nominative and accusative.

Both can be possible, but they are used a little differently.

• etwas suchen = to look for something
• nach etwas suchen = also to look for something

In everyday German, etwas suchen is very common and natural:

• Mein Bruder sucht das Bügeleisen.

Using nach is also possible:

• Mein Bruder sucht nach dem Bügeleisen.

With nach, the noun changes to the dative:

• dem Bügeleisen

So the original sentence is completely normal.

Because weil introduces a subordinate clause. In German, subordinate clauses are normally separated from the main clause by a comma.

• Mein Bruder sucht das Bügeleisen, weil ...

This is a very standard rule in written German.

Weil is a subordinating conjunction, and in German subordinate clauses, the conjugated verb goes to the end.

Main clause:

• Mein Bruder sucht das Bügeleisen.

Subordinate clause:

• ..., weil er sein Hemd noch bügeln will.

The verb idea is split here because of the modal verb:

• bügeln = infinitive
• will = conjugated modal verb

In a weil clause, the verb material moves to the end.

Because this is a subordinate clause.

In a normal main clause with a modal verb, you would say:

• Er will sein Hemd noch bügeln.

But after weil, the verb goes to the end, and with a modal verb the usual order is:

• ..., weil er sein Hemd noch bügeln will.

So:

• main clause: Er will bügeln
• subordinate clause: ..., weil er bügeln will

Here noch means something like still or yet.

• sein Hemd noch bügeln will = he still wants to iron his shirt / he wants to iron his shirt yet

It suggests that the ironing has not happened yet, but he still intends to do it.

Because Hemd is a neuter noun: das Hemd.

The possessive word sein changes depending on gender, case, and number.

Here Hemd is:

• neuter
• singular
• accusative (because it is the thing being ironed)

For a neuter singular accusative noun, the form is:

• sein Hemd

Compare:

• sein Hemd = his shirt
• seinen Mantel = his coat (Mantel is masculine)
• seine Jacke = his jacket (Jacke is feminine)

Yes, in this sentence er naturally refers to mein Bruder.

• Mein Bruder is masculine singular
• er matches it

So the meaning is:

• My brother is looking for the iron because he still wants to iron his shirt.

Because all German nouns are capitalized.

So in this sentence:

• Bruder
• Bügeleisen
• Hemd

are capitalized simply because they are nouns. This is a basic spelling rule in German.

Will comes from wollen and means wants to.

• er ... bügeln will = he wants to iron

Möchte is often softer and more polite, more like:

• would like to

So:

• weil er sein Hemd noch bügeln will = because he wants to iron his shirt
• weil er sein Hemd noch bügeln möchte = because he would like to iron his shirt

The original sentence sounds direct and natural.

Yes. They come from the same basic idea.

• bügeln = to iron
• das Bügeleisen = iron, literally the ironing device/tool

This kind of word family is very common in German and can help with vocabulary building.