Any trader can create an offer with another trader, there are a few ways to do this. You can start from scratch on the "My Offers" page by using the "Create" link there, or you can use links in a User page header, from a mini-matches page, or from your "My Matches" page. It all ends up in the same place, the offer creation page, but some of these fill in some information for you (like who and what's in the offer) to make things easier for you.
While creating and offer, you can use the "Pick Listings" buttons to add or remove items from the offer, or if the items are not listed, you can just fill them in by hand in the free-form boxes. You also get to set when the offer expires, who sends when, shipping, and whether you want to pre-accept the offer. You can control what these default to on your "Settings" page.
Once you create an offer, it gets put in your "OutBox" (a folder link on the "My Offers" sub-menu). It also gets put in the other trader's "InBox", and they will generally get an email about it (though, Offer Email Notification can be disabled on the "My Account" page). A new offer also have an expiration date on it, if it does not get a response before the expiration, it expires and the offer is moved to the "History" folder, linked on your "My Offers" sub-menu.
Hopefully, the other trade will respond as that's the polite thing to do. We encourage everyone to respond to offers promptly so we're not waiting for each other's responses (even if it's just to quickly decline it). A response to an offer consists of either an acceptance (where the other trader agrees to the contents of the offer and a pending trade is created), a decline (in which the offer is refused and is simply moved to the History folder), or a counter-offer (in which the other trader can change the terms of the offer and send it back to the other trader). With any response, you can add a message to the offer if you like, if you want to say something to the other trader.
Counter-offers are the most interesting case. You can go back and forth as often as you need to, each trader making counter-offers, each counter-offer making adjustments to the contents of an offer. Adjustments include what is being traded, who sends first, and what sort of shipping is used. Plus, you can exchange messages this way. If it goes on long enough, you might have to extend the expiration date for the offer. Hopefully, this back and forth negotiation will eventually lead to both sides agreeing to the offer.Here's a breakdown of individual functions:
Items that are part of an active offer (an active is any offer which is not expired, agreed upon, or declined) cannot be editing or deleted from your "My Items" page. This is done to ensure that the description and condition of the items are not changed in the middle of negotiations.
You can control whether the Offer System emails you about Offers on your "My Account" page. See "Offer Notification:" under Email Alerts.
There are convenient links to the Offers system from User page headers and from the My Matches page ("Offer" link).