Before switching to the current writing system (generally known as chữ quốc ngữ), Vietnamese was written with Chinese characters in a system called chữ nôm. These were characters very much like what you see in Chinese, but with some adaptation for the way Vietnamese is pronounced. For example 三 is a character meaning "three", which should have a pronunciation similar to sam (edit: that is, back in the day when it was taken in). Howerver in Vietnamese the number 3 was pronounced more like ba, so the chữ nôm character was 𠀧, where 巴 represents the pronunciation and 三 represents the meaning.

All of this was just for writing the spoken (Vietnamese) language. Formal documents would have been written in Classical Chinese, where you'd still use 三. Classical Chinese was the lingua franca for official business throughout eastern Asia, but then different countries/languages had their own version of chinese characters which they'd use for less formal writings. Chữ nôm was the Vietnamese version. Gugyeol was the Korean version. I'm oversimplifying a little but you get the idea.

In 1910 they officially officially switched to the alphabet they have now under the influence of Missionaries and a desire to "modernise". In 1918 the emperor officially declare the character-based system done with, and a decade or so later the characters had finally truly died out.

Let me know if you need any clarification. There is also this thread from a month ago with some more information.