Vietnamese was formerly written with Hanzi, or Chinese Characters. Unfortunately, it didn't suit the spoken language particularly well, in part because Vietnamese is an Austroasiatic language while Chinese it Tibeto-Burman. This meant that it was hard to spread literacy amongst the middle class because learning to read and write required learning Classical Chinese first.
Other languages that were in the same situations, such as Korean and Japanese, developed alternative writing systems to better express their language in writing. Korean scholars developed hangul in the 15th century, for example. Korean society experienced increasing literacy rates until the Imjin Wars of 1592-1597, after which literacy levels dropped precipitously, and did not begin to rise again until 1905.
Vietnamese went the other way, initially. In the 13th century, Chinese-style characters that didn't already exist in Chinese were developed solely for Vietnamese words. These are called chu nom and they are actually quite interesting. Vietnamese poetry experienced a golden age once these new characters were well developed, and a great deal of Vietnamese literary culture up to the present day owes a debt of gratitude to chu nom and their early adopters.
The problem was that you still had to learn Chinese first if you wanted to become literate. So the average person lived their life in a state of illiteracy or semi-literacy for most of recorded Vietnamese history. When the Jesuits came to Vietnam in the 17th century, they set about trying to educate young Vietnamese people. This is a controversial topic, and I want to emphasize that I am not making a judgement on anyone or any institution before I proceed. In an effort to ease pedagogy, the Catholics developed quoc ngu, a romanization of Vietnamese. When the French conquered Vietnam, quoc ngu systematically replaced Chu Nom as the written language for everybody.
sunxiaohu has provided a very good answer. But if you are wanting more replies and detail, you might also like to ask this question in /r/asklinguistics.
Burma has its own script that fits its language much like Thai, Lao and Khmer. People in Hong Kong speak mainly Cantonese but Mandarin is also used widely now. As such the languages (not dialects since they diverged long ago much like French, Italian, Portuguese, Romanian and Spanish did from Latin) may use the same script.
Vietnamese is believed to have also had a similar script to that of Thai/Khmer but was lost due to the Chinese exposure. It's given the name Chữ Khoa Đẩu (tadpole script, 𡨸蝌蚪).
The script in usage today is called Chữ Quốc Ngữ (national language script, 𡨸國語) and began as missionaries required a way to transcribe works into a more easily readable script.
Chữ (Hán) Nôm [Chinese-Vietnamese character script, 𡨸(漢)喃] was used by many elites and used heavily in native Vietnamese poetry. As Vietnamese vocabulary was already half Chinese-derived and also tonal + had similar sounds to those of the southern Chinese varieties, they adapted upon the script and coined thousands of new characters for Vietnamese using similar methods employed by the Chinese.
Some examples include:
The word "sky, heaven" which today is either trời or giời, used to be tlời or blời. As it's phonetically dissimilar to anything Chinese had they coined a new character using 2 semantic (meaningful) components. 𡗶 is composed of 天 (heaven) atop 上 (upwards, above). Another example is𠆳 trùm meaning "chief". It is comprised of 人 (person) atop 上 (upwards, above).
A second way of coining characters is through removing components. This one is much rarer and a few example include: 𧘇 ấy using y 衣 but then removing the top component. 爫 làm takes 濫 and removes most of the right hand side.
A third way is to use the most commonly used method in Chinese, taking a phonetic (sound) and semantic (meaningful) component and meshing them together. Note that sometimes these were "re-coined" as the characters themselves had already existed in Chinese but were re-invented for this particular purpose. 咹 this character already existed but was re-invented to mean "eat" (ăn) taking 口 (mouth) + 安 (the sound "an"). Another variant is 𩛖 which takes 食 (food) + 安 (the sound "an"). 粓 (cooked rice) cơm which again existed is re-invented through 米 (rice) + 甘 (the sound "cam"). Nơi meaning "place" takes 土 (earth) + 尼 (the sound "ni").
The fourth way is to borrow a character for its sound only. The word "một" (one) borrows 没 regardless of its extant meaning "not have, sinking".
The fifth way is simply to write down the words which were formally loaned over but read in a Vietnamese manner. These Sino-Vietnamese words are the most common and account for about 50% of the vocabulary. Words pertaining to education (教育, giáo dục), governance (政治, chính trị), the sciences (科學, khoa học) etc. are the most commonly borrowed in this category (though the word for "science/s" was borrowed via Japanese coinage much like that of "telephone, điện thoại (電話)" which was originally coined in Japan > borrowed into Chinese > borrowed into Korean and Vietnamese).