Commas with "as"

As you've pointed out, the word as is sometimes used to connect two clauses. When as is used this way, it generally means while, when, or because, depending on the context, and it is called a subordinating conjunction. The rules about when to use a comma with as are the same as the rules for using commas with other subjordinating conjunctions, including while, when and because. Here they are: 

 If the clause with as comes first, a comma is generally required.

• As the plane was taking off, Erica fell asleep. (as =while)
• As I'm a pacifist, I'm against all wars. (as =because)

If the clause with as comes second, a comma is generally not required.

• Jonas fell as he stepped off the curb. (as =when)
• She stayed home as she had no car. (as =because)

However, even in a sentence in which the "as clause" comes second, a comma may be necessary to clarify the meaning. 

• She went home as it was raining. (could mean either because it was raining or while it was raining)
• She went home, as it was raining. (clearly means because it was raining)