chiếc

The standard word order is: (Number) + chiếc + Noun + (Adjective) + (Demonstrative). If there is an adjective, it goes between the noun and the demonstrative (e.g., một chiếc áo mới này).

While cái is the general classifier for inanimate things, chiếc is standard for certain categories. It can also replace cái for many physical objects to sound more formal, literary, or poetic. Except for modern internet slang, chiếc is strictly used for inanimate objects. Do not use it for animals or people (which require classifiers like con or người). Furthermore, unlike cái, which can turn adjectives into abstract nouns (like cái đẹp for "beauty"), chiếc is generally used for physical, tangible objects, though you may occasionally see it used poetically with non-tangible things like chiếc bóng (a shadow) or một chiếc hôn (a kiss).

Crucially, in formal grammar, a classifier (like chiếc or cái) is generally expected when you want to point out a specific item using a demonstrative like này (this), đó (that), or kia (that over there), such as in chiếc áo này (this shirt). However, in everyday speech, native speakers frequently drop the classifier (e.g., áo này).

Here is when you will most commonly see it used:

• Vehicles: chiếc xe máy (motorbike), chiếc ô tô (car), chiếc thuyền (boat), chiếc máy bay (airplane). Note that with vehicles, dropping the classifier is actually very common and perfectly grammatical (e.g., hai xe máy).
• One of a pair: Because chiếc literally translates to "single" or "solitary", it is strongly preferred for one item of a pair, like một chiếc đũa (one chopstick) or một chiếc giày (one shoe). To refer to the whole pair, you should use the classifier đôi instead (e.g., một đôi giày).
• Clothing & Accessories: chiếc áo (shirt), chiếc nhẫn (ring). Using chiếc instead of cái here sounds slightly more elegant.
• Furniture & Household Items: chiếc bàn (table), chiếc ghế (chair). Similar to clothing, using chiếc instead of cái for these items adds a touch of elegance.
• Flat or Delicate Objects: chiếc lá (leaf), chiếc nón (conical hat).

Important Notes:
For strong emphasis, you might hear the 'super-classifier' cái placed directly in front of chiếc (e.g., cái chiếc xe này - this very car). Additionally, in modern internet slang, young people sometimes use chiếc with unexpected nouns (like một chiếc mèo for "a cat") to sound cute or endearing, but you should stick to the standard rules for everyday speaking!