Wait for bids. Once someone makes a bid, acknowledge their bid by repeating it with their name. I have 1g on [Iridescent Pearl] by Jonny. If people are bidding really quickly against each other, wait until it slows down and announce the last bid.

If bidding slows down or stops, wait a bit (it's up to you how long) and say [Iridescent Pearl] going once for 1.5g to Jonny. (or whatever the last bid was). Wait a little bit longer then say [Iridescent Pearl] going twice for 1.5g to Jonny. Then call the auction as sold. [Iridescent Pearl] sold for 1.5g to Jonny.

It's important that you keep repeating the link and the price so people who come into the channel later can see what is being bid on and what the price is at. It's also important that you call the auction. Otherwise the buyer might become bored and will quit.

You should now message the winning bidder and tell them that they won and set up a place to meet. Sellers should locate themselves by banks or flying transport locations so that buyers can easily find them.

If the deal falls through somehow and you're unable to work out the problems with the winning bidder, offer the item up for auction again. You can also try selling it to the second highest bidder at their last bid price. However, they are under no obligation to pay since the auction is over.
Other tips

Don't try to sell an item that cannot be traded such as a soulbound item. Read about item basics to learn about the quality of items and bind status. Many quest items cannot be traded. Check to see.

Buyers
As items scroll by, make a bid on an item you want. Example: "10 s (silver) on [Amazing Sword]." Try to mention what you are bidding on if there are multiple items for trade at the same time.
Beware of bid wars
The goal of the auction isn't to BEAT another player in some imaginary battle. The goal is to get the item at a reasonable price. Don't bid up an item to crazy numbers just to win against another person bidding on the item. Pull out if the price gets too high.