飾｛かざ｝り, デコレーション, 飾｛かざ｝り付｛つ｝け, and 装飾｛そうしょく｝
Before distinguishing these words, I looked up "decoration" in several English-English dictionaries, and I found that there are roughly three meanings as follows:
(1) The act, process, technique, or art of decorating
(2) Something used to decorate, especially when put up temporarily to celebrate or call attention to a special occasion
(3) An emblem of honor, such as a medal or badge
At first I can say that only "飾り付け" is different from the others; it has the meaning corresponing to (1) in the definition of a dictionary; that is also said in Seesawscene's answer.
As for "デコレーション", "飾り" and "装飾", they have nothing to do with the meaning of (3).
I also judged them by google images (without "クリスマス" added), and found they're something like this.
- bright, colorful, showy ---- デコレーション
- imposing, monolithic, massive, dignified, stately ---- 装飾
- N/A ---- 飾り
- temporary ---- デコレーション、飾り
- permanent ---- 装飾
Judging from this classification, "装飾" may not correspond to (2) according to the definition of the dictionary. As an expression to support this, we say "装飾を施｛ほどこ｝す lit. to operate/apply 装飾", while we say "飾りを付ける lit. to attach 飾り"
- Western style or modern ---- デコレーション
- Japanese style or traditional ---- 飾り
- N/A ---- 装飾
Although the application to each classification is my personal judgment, I think it is not so deviated from the general judgment.
As a fact that supports these decisions, I'll show below where these words came from.
- A loan word from English ---- デコレーション
- A loan word from Chinese ---- 装飾
- A Japan's own word ---- 飾り