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). You may have heard that the Chinese particle 着 added onto the end of verbs is similar to the use of -ing in English. This isn't particularly helpful, however, because the use of 着 in Chinese is not nearly so frequent, and can also be quite idiomatic.

Basic Structure

Examples

Zhè jiā fàn diàn hái kāi zhe.
这 家 饭店 还 开 着 。
This restaurant is still open.

着 for a continuous state
While it's true that the "full progressive pattern" can make use of 着, this is not a pattern you're going to want to use all the time. For example, if you want to say "I'm reading," you have these two choices:
√ 我 在 看 书 。(This is the natural, easy way to say it.)
× 我 正 在 看 着 书 呢 。 (This seems a bit much, and isn't very natural.)
The first one is fine, but the second one is definitely odd, and unnecessarily wordy. So there's no need to intentionally construct such long, unwieldy structures. For this kind of usage (which corresponds pretty closely to the "-ing" in English which we mentioned before), you're better off avoiding 着.
There are, however, other uses of 着 which are needed. When you're talking about "states" which don't involve any continuous action, or actually doing anything, you're going to want to use 着 instead of 在. Some examples:

着 Expressing an Ongoing State

Verb + 着 Explanation
开 着 开 alone can mean "to open" or "to turn on." Adding 着 allows one to express that something "is open" or "is on.".
关 着 关 alone can mean "to close" or "to turn off." Adding 着 allows one to express that something "is closed" or "is off."
带 着 带 alone means "to carry." Adding 着 allows one to express that one "is carrying" or "has" something (on one's person).
坐 着 坐 alone means "to sit." Adding 着 allows one to express that someone "is sitting" ("在坐" is awkward, because it's not a real action).
躺 着 躺 alone means "to lie on one's back." Adding 着 allows one to express that someone "is lying down."

Now let's see these in action:
√ 房间 里 灯 关 着 。 ("Being off" is a state, so using 着 is natural.)
× 房间 里 灯 在 关 。 ("Being off" is not an action, so don't use 在.)

着 for doing an action in a particular state
If you do an action while in a particular state, you can make use of this pattern:
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.
站 着 吃饭 "standing + eat = eating while standing"

Note: If you want to make a sentence where both verbs are action verbs (neither is truly a state), then you don't want this pattern, you want 一边⋯⋯，一边⋯⋯.

着 Used Idiomatically
Certain verbs tend to take 着 more frequently than others, and exactly what the 着 is doing might not be apparent at all. It's best to think of these usages as colloquialisms. You can even think of them as set phrases.
着 Used Idiomatically

Verb + 着 Explanation Example
听 着 "to listen to" (essentially the same as 听) 你 听 着。
拿 着 "to hold" (essentially the same as 拿) 这个 你 拿 着。
等 着 "to wait" (essentially the same as 等) 你们 继续 等 着 ！

There's also one colloquial usage of 着 that's been chosen by at least one textbook for special treatment, so we'll cover it here as well:

This pattern may look like that "doing an action in a particular state" pattern already covered above, but in practice it doesn't really work that way. It just means "[Verb] for fun" or "[Verb] as a joke."

Examples of this usage:

Wǒ bú shì rèn zhēn de, wǒ shì nào zhe wán ér .
我 不 是 认真 的，我 是 闹 着 玩儿。
I'm not very serious, I like to have a lot of fun.