What are the different kinds of poems in Chinese?
At the beginning, I recommend you read the articles on Wikipedia to get some background info, because my English level limits me to elaborate on these info.


诗 Shi;
词 Ci;
曲 Qu (here it refers to 歌 in the question);
赋 Fu;


I am quite not familiar with the poetry in English or some other languages. Here I would only like to explain how I think a Chinese poem is defined, and I can't tell the differences between chinese poems and poems in other languages.
As the language changed, the style or pattern of Chinese poetry also changed. Nowadays, most famous and popular classic Chinese poems are 唐诗, the poems written in the 唐 Dynasty. Here, I would like just to explain the style of 唐诗, because they are the most popular and the style definition is strict. Remember, there are other styles of poems, but I won't talk about them here (too complicated, sorry).
In most cases, 唐诗 is subject to rules called 格律 (gé lǜ). 格律 includes four aspects: 用韵, 对仗, 平仄 and 字数.

字数: number of characters
五言绝句: 4 sentences with 5 characters for each sentence.
七言绝句: 4 sentences with 7 characters for each sentence.
五言律诗: 8 sentences with 5 characters for each sentence.
七言律诗: 8 sentences with 7 characters for each sentence.

用韵: to rhyme
The last character of the sentences, if its tone is 平 (see below), it should rhyme. I will show it in examples below. Note: Because pronunciation in those times was different from that in modern mandarin, you would find some sentence don't rhyme in mandarin.

对仗: to match, to relate to, to pair
It's hard to explain, I will show it in the examples. This characteristic is optional for 绝句, but required for 律诗.

平仄 (píng  zè)
平仄 refers to the tone of the characters. In ancient times (before the 元 dynasty), Chinese had 4 basic tones: 平, 上(shăng, not shàng), 去, 入. However, 入 has disappeared in mandarin, while in some dialects, especially in southern dialects, it remains. 上, 去 and 入 are classified into 仄. For 唐诗, the tone of every character is asked to satisfy a defined pattern based on 平仄.


Ok, now it's time for the examples. Thereinafter, &quot;-&quot; means the tone should be  平，“|” means tone should be 仄, and &quot;+&quot; means the tone could be either 平 or 仄.

A 七言绝句 from 杜甫


+-+|--|，+|--||-。+|+--||，+-+||--。
两个黄鹂鸣翠柳，一行白鹭上青天(tiān)。
窗含西岭千秋雪，门泊东吴万里船(chuán)。

In this example, 天 and 船 have 平 tone, and they rhyme (containing the same consonant &quot;an&quot;). Furthermore, 对仗 happens between the 1st and the 2nd sentence, and also between the 3rd and the 4th sentence. Read the 1st and the 2nd sentences (I wish you could), you would find what 对仗 means.

两 means &quot;two&quot; while 一 means &quot;one&quot; (both of them are numbers). 黄 means &quot;yellow&quot; while 白 means &quot;white&quot; (both of them are colors). 翠 means green while 青 means blue(again, colors). 个 and 行(háng) are all measure words. 黄鹂 and 白鹭 are names of two kinds of birds. 鸣 means &quot;[a bird] to cry&quot; and 上 means &quot;to soar, to fly high&quot;. Both of them are verbs. 柳 means willow, and 天 means sky. Both of them are nouns.

This rhetoric is called 对仗: words at the same position have similar or reverse or related meanings(depending on the context), and have the same part of speech.
词 somehow works like 诗. The difference is that 词 has many different styles, called 词牌. One example would help me explain it: It's a work by the famous female writer 李清照, with its 词牌 of 《如梦令》. Below, &quot;韵&quot; means this character should rhyme. &quot;叠&quot; means this sentence should repeat exactly its previous one.

+|+--|(韵),+|+--|(韵)。+||--，+||--|(韵)。-|(韵),-|(叠),+||--|(韵)。
昨夜雨疏风骤(zhòu)，浓睡不消残酒(jiŭ)。试问卷帘人，却道海棠依旧(jiù)。知否(fǒu)，知否？应(yīng)是绿肥红瘦(shòu)。

Another 《如梦令》also by her

常记溪亭日暮(mù)，沉醉不知归路(lù)。兴(xìng)尽晚回舟，误入藕花深处(chù)。争渡(dù)，争渡。惊起一滩鸥鹭(lù)。

A work by the famous writer and general 辛弃疾, with 词牌 of 《南乡子》

+||--(韵)，+|--||-(韵)。+|+--||，--(韵)，+|--+|-(韵)。
+||--(韵)，+|--||-(韵)。+|+--||，--(韵)，+|--+|-(韵)。
何处望神州(zhōu)，满眼风光北固楼(lóu)。千古兴(xīng)亡多少事，悠悠(yōu)，不尽(jìn)长江滚滚流(liú)。
少年万兜(dōu)鍪(móu)，坐断东南战未休(xiū)。天下英雄谁敌手？曹刘(liú)，生子当如孙仲谋(móu)。

曲 is very similar with 词, having a lot of different styles, called 曲牌. It was originally composed for operas and was popular in 元 dynasty.
赋 is more like a prose, so it would be much longer, compared to 词 and 曲. It was popular in Han dynasty and before that time.
As I shown, when composing a classic 诗 or 词, the writer should pay attention to these rules, especially the tones of 平仄; however, it's hard for a man who could only speak in mandarin or some dialects to tell 平仄 of a character. Besides, limited by the character numbers, the writer has to express the meanings in a classic way, with few characters. This is hard too. Nowadays, it's very very rare to see a classic 诗.
Based on the descriptions above, you may understand why 《三字经》is not treated as 诗 by Chinese people, though the sentences rhyme.