Learn Arabic (Modern Standard): هذا الكتاب جديد.

Arabic doesn't use a word for "is" or "are" in present tense sentences like this one. You just put the subject (هذا الكتاب, "this book") next to the information about it (جديد, "new"), and the "is" is naturally understood.

Why does the word for book (الكتاب) have "al-" (the) attached to it? Doesn't that mean "This the book"?

In Arabic, to say "this book", the noun following "this" (هذا) must have the definite article الـ (al-). If you left it off and said هذا كتاب (without "al-"), it would change the meaning completely to "This is a book."

Why doesn't the word for new (جديد) also have "al-" (the) attached to it?

Because جديد (new) is the news or information about the book, rather than an adjective just describing it. If you added "al-" to make it هذا الكتاب الجديد, you would be saying the incomplete phrase "this new book" instead of "This book is new."

I notice the word هذا (this) sounds like it has a long "a" after the first letter, but I don't see an alif. Is that a typo?

It is not a typo! The word هذا is one of a few special words in Arabic that has a "hidden" long "a" sound. It is pronounced "haadhaa", even though the first alif is not written out in standard spelling.