Spoken Mandarin Chinese comprises a certain small number of initial and final sounds, 21 initials and 16 finals to be exact. All syllables, and words, are made up of a combination of these. All the syllables start with an initial sound, which is a consonant, and end with a final sound, which are the vowels. The vowels can combine among themselves to make compound vowels, for instance "i" and the nasal vowel "an" to form "ian". When we add the initial and final sounds we get 56 basic sounds, all the combinations among them yield 413 possible syllables. But of course, Chinese people have more than 413 words! This is because the syllables are spoken with one of the four tones that exist in Chinese. Combining the sounds with the tones yields 1600 unique syllables1.
Phonetic symbols are used to write the Chinese sounds. The most well known phonetic system is pinyin, which literally means "phonetics". It is so famous because the sounds are represented with roman characters. But pinyin is not the only system. Another well known system is called zhuyin, or bopomofo (the name originates in the sound of the first 4 consonants b-p-m-f). Zhuyin is a Chinese system and does not use letters; it has its own set of simple zhuyin characters. There is a third system called Wade-Giles after the names of the inventors, also based on roman characters.
Let's see examples of pinyin and zhuyin systems. Here is the Chinese word for "good".

But, how is it pronounced? It is difficult to know this without a phonetic key, since the characters don't say much about its pronunciation. The initial sound for this word in pinyin is symbolized with the roman letter "h", which has a sound similar to the English "h" in the word "holiday". In zhuyin system, this sound is represented with a simple symbol "
". The initial consonant is followed a final vowel sound represented as "ao" in Pinyin. It is very similar to the "ow" sound in the English word "owl". The zhuyin representation is "
".
The phonetic writing would be "hao" in Pinyin, and "
" in zhuyin.
But the last detail to be added to the phonetic key is the tone. There are four tones in Chinese. "Hao" meaning good is pronounced with the third tone, which is a wavy tone. At first it has a normal pitch, then it descends, and then ascends again. This wavy tone is represented very properly with a wave symbol in both pinyin and zhuyin: 
Therefore, the complete phonetic writing of the word is "hǎo" in Pinyin, and "
" in zhuyin. Zhuyin characters are usually written vertically, though.
Chinese children start learning how to read and write at ages 6 or 7, and at first, it is as difficult for them to process the complex characters as it is for an English speaker. So, they also start learning the few phonetic symbols and then move on to the thousands of characters. Let's see another example:

hǎo (good)

hěn (very)
Traditionally, the system to teach children was zhuyin. Recently, it's been officially replaced with the pinyin, except in Taiwan, where you might even find children's books with the zhuyin annotations next to the characters.
The Wade-Giles system was widely used in the West to write Chinese names before pinyin. But, since it is more difficult to learn and a bit less accurate, it is now also less common. Very nice examples of the Wade-Giles system are the names of Chinese cities. Peking, Canton and Nanching have been recently replaced with the pinyin names Beijing, Guangzhou and Nanjing.
You can now read and understand the following Chinese expression, used everyday when greeting someone:

!
1. Complete list of basic initial and final sounds in pinyin:
Initials |
Unaspirated |
Aspirated |
Nasal |
Voiceless fricative |
Voiced fricative |
Labial |
b |
p |
m |
f |
|
Alveolar |
d |
t |
n |
|
l |
Velar |
g |
k |
|
h |
|
Palatal |
j |
q |
|
x |
|
Dental sibilant |
z |
c |
|
s |
|
Retroflex |
zh |
ch |
|
sh |
r |
6 simple final - a, e, i, o, u, ü
13 compound finals - ai, ao, ei, ia, iao, ie, iou, ou, ua, uai, üe, uei, uo
16 nasal finals - 8 front nasals: an, en, ian, in, uan, üan, uen, ün
8 back nasals: ang, eng, iang, ing, iong, ong, uang, ueng