You never bring uninvited guests.
If you're single, the invitation should just say "James". If you are dating someone and the couple has chosen to invite her, she should receive her own invitation mailed to her address. You might get an invitation that says "James and Guest" in which case it's ok to bring a date (but this isn't "proper" - it's preferable that the couple find out your date's name and address and send an invitation directly to her). Otherwise, she's not invited, and you don't get to just show up with her.
If you're engaged, the couple has to invite both of you. If you and she are not living together, you should each get separate invitations as described above. If you are living together, you should receive one invitation with both of your names. It would look like:
Ms. James
Mr. James
Your Address
City, State, Zip
If you are living together but not engaged, it is treated the same as if you were engaged and living together.
If you're married, again the couple has to invite both of you. The invitation would be addressed like:
Mr. and Mrs. James
Your Address
City, State, Zip
or, if you have different last names:
Ms. Hername and Mr. James
Your Address
City, State, Zip
Children are invited if and only if their names appear on the invitation:
Mr. and Mrs. James
James Jr.
Your Address
City, State, Zip
If the kids aren't listed, they aren't invited.
(Yeah, I spent way too much time on alt.wedding a few years back.
)