What are other popular composition techniques in addition to The Rule of Thirds?

Question

I always hear a lot about The Rule of Thirds. I'd like to know more about other 'tried-and-true' composition techniques (not special effects) that can make a photo more interesting.

In particular, I'd especially like to know:

• The name of the technique
• Any particular types of settings the technique is particulary useful
• Interesting ways to 'break' the rule

Answer

While this isn't a duplicate, this can essentially be answered by linking to a few questions we've collected regarding other composition techniques (thanks largely to @JayLancePhotography!):

• Bakker's Saddle
• Rule of Odds
• Diagonal Method
• Golden Ratio

Searching the composition and composition-basics tag provides a wealth of knowledge.

Check more discussion of this question.