الجاكت ده من الصوف.

Because in Egyptian Arabic, this/that usually comes after the noun.

So:

• الجاكت ده = this jacket
• literally: the-jacket this

This is different from English, and also different from Standard Arabic, where you usually get:

• هذا الجاكت or more naturally هذه السترة = this jacket

In Egyptian Arabic, noun + ده/دي is the normal pattern.

ده is the Egyptian Arabic masculine singular demonstrative, meaning this.

In everyday Egyptian:

• ده = this for masculine nouns
• دي = this for feminine nouns
• دول = these

So in الجاكت ده, ده agrees with الجاكت as a masculine singular noun.

You may also see it written دا. Both ده and دا are used to represent the same spoken Egyptian word.

Because Arabic often uses a nominal sentence in the present tense, with no separate word for is/am/are.

So:

• الجاكت ده من الصوف
• literally: this jacket from wool

But the natural English translation is:

• This jacket is made of wool
• or This jacket is wool

The verb to be is usually omitted in present-tense Arabic sentences like this.

الـ is the definite article, meaning the.

So:

• جاكت = a jacket / jacket
• الجاكت = the jacket

In Egyptian Arabic, when you use noun + ده/دي for this/that, the noun is normally definite, so it usually takes الـ.

That is why you say:

• الجاكت ده = this jacket

not normally just:

• جاكت ده

In Arabic, materials and generic substances are often expressed with the definite article.

So:

• من الصوف = made of wool
• literally: from the wool

This sounds natural in Arabic even though English usually says just of wool without the.

You can think of الصوف here as referring to the material wool in a general sense, not a specific piece of wool.

من often means from, but in sentences about material, it commonly means made of.

So here:

• من الصوف = of wool / made of wool

Other examples:

• الترابيزة دي من الخشب = This table is made of wood
• الخاتم ده من دهب = This ring is made of gold

So من has a wider range than just physical from.

It is a borrowed word, ultimately from jacket.

Egyptian Arabic uses many everyday loanwords, especially for clothing and modern objects. So الجاكت is perfectly normal colloquial Egyptian.

You may also hear or see variations such as:

• جاكت
• جاكيت

Spelling can vary because colloquial Egyptian is less standardized in writing than Standard Arabic.

In this sentence, it is treated as masculine, because it takes ده.

So:

• الجاكت ده = masculine

With many borrowed nouns in Egyptian Arabic, gender has to be learned from usage. Here, the sentence itself tells you the gender because the demonstrative is masculine.

A common Egyptian pronunciation would be roughly:

• il-gākit da min is-sūf
• or el-gaket da men es-soof

A few notes:

• الـ is often pronounced il-/el-
• ده is pronounced like da in speech
• من is often pronounced min or men
• الصوف is pronounced is-sūf / es-soof, because the ل in الـ assimilates to the ص

So the written form has ال, but the spoken form sounds more like es-soof.

Because ص is a sun letter.

In Arabic, when الـ comes before a sun letter, the ل sound is not pronounced, and the following consonant is emphasized instead.

So:

• written: الصوف
• pronounced: as-sūf / is-sūf / es-soof depending on the variety and how you represent it

This is a very common Arabic pronunciation rule.

Yes, it sounds natural and straightforward.

الجاكت ده من الصوف is a normal Egyptian way to say:

• This jacket is made of wool

A speaker might also say slightly different versions depending on style, such as:

• الجاكت ده معمول من الصوف = This jacket is made of wool
• الجاكت ده صوف = This jacket is wool/woolen in some contexts

But your original sentence is perfectly good everyday Egyptian.