your sentence would be: هذه الشنطة ثقيلة
The change in the definition of the word (i.e. having AL or not having AL) changes the meaning. The sentence above, the adjective (heavy) is attributive to (bag), that is, it describes the bag.
In your sentence, you're informing the listener that this particular bag is heavy.
Notice the difference in the position of "is" in the sentence above and your sentence; This is the marker for which part is the predicative of the sentence (i.e. the information that the sentence is trying to tell and determine). What comes after "IS" (or the verb "to be" in general) in English is usually the predicative, and what comes before it is usually the subject. In Arabic, this is all controlled by changing the definition of the word with "AL". The predicative is typically indefinite (without AL) in Arabic.