свой refers back to the subject of the sentence. If the subject is "I" then свой means "my", if the subject is "he" then свой means "his" and so on.
свой and своего are equivalent to мой and моего (or твой and твоего). So свой is the normal masculine form, and своего is the genitive masculine form, and also the accusative masculine form when the object is animate.
The question remains unanswered. According to http://www.russianlessons.net/grammar/pronouns.php, at "Reflexive possessive pronoun 'Свой,'" the table shows masculine Accusative Case (animate) as Свой or Своего.
Your question is how свой differs from мой (these are nominative). Свой is "one's" and Мой is "my." We don't use "one's" in (American) English much because... well... one doesn't talk like that. One's vocabulary doesn't utilize it a lot. (Works like that.)
You also asked whether своего or моего (the Accusative or Genitive forms) could be used. Keep in mind the Tips & Notes from the "Family" section:
Unlike English, Russians rarely say "my mother", "my grandfather"; usually they omit "my".
...And when they don't, it is more natural to use reflexive "свой" (one's own).
In response to another user on this same thread, своего is not "us."
(Edited because for some reason, something I did with formatting gutted a couple of paragraphs.)
No matter what i do it still makes me feel like the person who is loved belongs to somebody else :)
So possessives match the possessee. The reason it's мужа and not just муж is because the word is masculine and animate, so the accusative form is the same as the genitive (мужа) instead of the nominative (муж).
The same is then true for the possessive. So if we're using a form of мой, then мой is the nominative form and моего is the genitive form, so the accusative form is also моего and we get моего мужа. мой мужа is incorrect for these reasons.
Actually, there's an interesting rule that only applies to masculine nouns. If the noun is animate (i.e. a person or animal), then the accusative form is the same as the genitive. Otherwise the accusative form is the same as the nominative.
That's why мужа matches the genitive case. The same also applies to determiners/adjectives of masculine nouns. That's why in this sentence своего is used instead of свой.