How does the particle word "呢" work? My teacher said that a particle word in Chinese is a function word, but don't quite understand what is means. How does the particle word such as "呢" work?

Also, are there any other Chinese particle words out there? Thanks in advance!
As others have pointed out, 呢 is a sentence-final particle that nowadays mostly has an interrogative function, namely to form a question regarding a topic that has already been brought up.

The normal interrogative particle is 吗, as in 你是中国人吗？ Or more rhetorically using 吧: 你是中国人吧？ But if you first introduce your own nationality, the question would instead involve the particle 呢: 我是丹麦人，你呢？

If you read older literature like 红楼梦, you will also find 呢 as a sentence-final particle that is not interrogative, but rather emphatic. It may indicate softening or suggestion, like in the sentences 尤氏的母亲已先在这里呢 and 太太们都在楼上坐着呢，请奶奶就从这边上去罢。

Other such particles are 吧 (softening of statements) and 啊 (affirmation).

Then there is a whole range of exclamative particles and interjections used in spoken Chinese, such as 呗 (when stating something obvious or with authority), 嗯 (acknowlegding), 哦 (surprise), and 哇 (awe).

Functional particles that are essential to Chinese grammar include 着 (durative aspect particle for continuous action, 我穿着衣服 = I am getting dressed), 了 (completion, 我回来了 = I am back, or 我不抽烟了 = I am not smoking anymore), and 的 (possessive particle: 我的车子).

Particles are thus used to modify words and sentences. They are not words per se, but add structure to sentences.