What you describe sounds simple enough, but what safeguards are there that the person isn't going to rip you off? Isn't it taking a really big chance to send a nameless, faceless person $100 and trust that they'll send you the bitcoin you're expecting?
Yes. You're also taking a big chance sending a nameless, faceless person 0.10 bitcoin and trusting that they'll send you the cash you are expecting. Now you are beginning to see why the process can seem difficult and complicated when the transaction isn't between two entities that already trust each other. localbitcoins reduces some of that risk with an escrow feature and peer rating.
Isn't there some sort of peer rating or something?
Yes. localbitcoins has a peer rating system. bitcointalk.org has a peer rating (trust) system. There is also the IRC #bitcoin-otc channel's WOT rating system.
And if I choose a seller who wants money deposited to a specific bank - how does that work? Do I go to any branch of that bank, say Wells Fargo, and deposit my cash into a bank account number the seller specifies?
Exactly.
Or if the purchase is made using a postal money order - don't they require filling out the names of the sender and receiver?
In my experience only the receiver's name and address are required. The sender does not need to supply any identifying information.
Thank you for taking the time to answer these questions. The answers can seem obvious to an experienced person, but to a person who has never made a transaction the answers are not always obvious.