How to use the grammar 着？

1、表示事物的存在：【V+着】：强调事物【某种状态的持续】。
e.g:墙上【挂着】一张画（A picture hangs on the wall）；
桌子上【放着】一本书（There is a book on the table）；
床上【躺着】一个孩子（There is a kid lying on the bed ）
2、表示动作的进行：【V+着】：强调【动作的进行】，也常常与“【正、在、正在】……”搭配使用。
e.g: 我们(正）【上着】课呢（We are having a lesson）；
他正【吃着】饭（He is eating）；
大家在【看着】电视呢（People are watching TV）

hsk2: 着（auxiliary）
1, Sub + V + 着（+数量）+ N
means that a state continues
例：他头上戴着一顶帽子。
2, V + 着（+N）
means that the action cotinues
例：她认真地听着，仔细地观察着。

1. It represents the present continuous tense. (eg. verb+"着"）
2. It represents the accompanying state. (eg. verb+"着"）

The particle 着 (zhe) is one way of indicating the continuous aspect in Mandarin Chinese (another common way is using the adverb 在 in front of verbs).
Structure: Verb + 着 (This basic pattern is often used with commands involving certain verbs where the action persists for a while)
example : 我们 做 ，你们 看 着 。Wǒmen zuò, nǐmen kàn zhe.We will do it, and you all watch.

Used for Manner or State in which an Action is Performed
Structure: Verb 1 + 着 + Verb 2
Note that the first verb (followed by 着) describes the state; the second verb is the action verb. In this case, the "-ing" translation can be useful for the state
example : 他 笑 着 说 “对不起” 。"Tā xiào zhe shuō "duìbuqǐ".Smiling, he said, "I'm sorry." (smiling + say = saying "I'm sorry" while smiling

Colloquial Sayings

Certain verbs tend to take 着 more frequently than others, and what the 着 exactly is doing might not be apparent at all. It's best to think of these usages as set phrases.
Examples : 听 着 ! Tīng zhe! ("to listen and keep listening")

着 is a particle. Verb + 着 means this action is in progress. 看着 looking\staring,吃着 eating, 坐着 sitting, 想着 thinking...