Vietnamese is a tonal language. By changing the pitch of your voice, you can completely change the meaning of a word.
For example, ba, said with a flat tone, means “three”. Pronounce it with a rising tone, as if you were asking a question in English, and it means “governor”. Pronounce it with a low, throaty tone and it means “randomly”.
Tones are hard for a native English speaker to master. A full explanation of how to pronounce them is beyond the scope of this article. For now, you need to know that Vietnamese has six different tones.
In many tonal languages (e.g. Chinese), the writing system doesn't provide much information about which tone to use for a given word. Or the tone might be encoded in the written word, but the rules for figuring it out are very tricky and convoluted (as in Thai).
Here's the good news: reading Vietnamese tones is very easy. This is because the tone is clearly denoted using one of five accent marks. There are five such symbols for the six tones, because the “flat” tone is denoted by having no accent mark. Four of the tone symbols are written above the vowel, and one is written below.
Again, I can't precisely explain in writing how the six different tones are pronounced. For now, just understand that they are pronounced differently.
Also note that in southern Vietnam, the “tumbling” tone isn't used. Southerners pronounce mã the same as mả, although they're still written differently.
So take the letter e, which as I've already explained is pronounced like in the English “get” or “bet”. Depending on the tone, this could be written e, é, è, ẻ, ẽ, or ẹ.
A syllable can't have more than one tone, so you'll never see two tone marks on the same letter. E.g. the hổi and ngã symbols ( ̉ and ~) would never be used together.
However, you can add a tone symbol to an “accented letter” like ă, ê, or ư. This is the only occasion on which a Vietnamese letter can have more than one accent. So ế, ỗ, ợ, and ữ are all valid, to give just a few examples.
Note that tone marks are only written on vowels. You'll never see ´ or ~ written above a consonant. But note that “y” is considered a vowel in Vietnamese, so ý, ỳ, ỷ, and ỹ are valid combinations.
So, in summary: there are two types of “accent marks” in Vietnamese. First, there are the accents on ă, â, ê, ô, and ư, which tell you that you're dealing with a different vowel from the equivalent with no accent. Second, there are the tone marks: é, è, ẻ, ẽ, ẹ, which can appear on any vowel, including the six vowels which already have some kind of accent mark. A vowel can have one accent from each category, as in ể or ộ – and this is the only circumstance in which a Vietnamese letter can have more than one accent mark.
Hopefully this long explanation has made things crystal-clear.