Three (三, formal writing: 叁, pinyin sān, Cantonese: saam) is considered a good number in Chinese culture because it sounds like the word “alive” (生 pinyin shēng, Cantonese: saang), compared to four (四, pinyin: sì, Cantonese: sei), which sounds like the word “death” (死 pinyin sǐ, Cantonese: sei2).
Counting to three is common in situations where a group of people wish to perform an action in synchrony: Now, on the count of three, everybody pull! Assuming the counter is proceeding at a uniform rate, the first two counts are necessary to establish the rate, and the count of “three” is predicted based on the timing of the “one” and “two” before it. Three is likely used instead of some other number because it requires the minimal amount counts while setting a rate.
In Vietnam, there is a superstition that considers it bad luck to take a photo with three people in it; it is professed that the person in the middle will die soon.
There is another superstition that it is unlucky to take a third light, that is, to be the third person to light a cigarette from the same match or lighter. This superstition is sometimes asserted to have originated among soldiers in the trenches of the First World War when a sniper might see the first light, take aim on the second and fire on the third.
The phrase “Third time’s the charm” refers to the superstition that after two failures in any endeavor, a third attempt is more likely to succeed. This is also sometimes seen in reverse, as in “third man [to do something, presumably forbidden] gets caught”.
Luck, especially bad luck, is often said to “come in threes”. (http://en.wikipedia.org/wiki/3_(number))