(taken with permission from a post on the forums by Hirokache)
Trade Post - Basic Information
Trading posts allow "Secure Trade". Two people standing nearby a Post click on the post and choose to trade with each other. A window (that looks quite out of place with the rest of Egypt) pops up, showing your inventory, a space for adding the items you wish to trade, and a space showing what the other person has added. Each person adds until they are happy and clicks agree, once both people have done so the trade is made. It is not possible to remove just one item from what you have selected to trade, so if you make a mistake you have to clear it all out. I personally only use Trade Posts if I am trading away some good on the behalf of others which I will not be able to replace if stolen, or I am making a complicated trade involving several items, as the Post makes doing so much easier.
Trade Center - Basic Information
Trade Centers have much in common with guild halls (and use the pre-live guildhall art). You apply to join a Trade Center and must be accepted. From then on though, you can leave and join almost at will (the command to do so is slipping my mind at the moment). Centers have size limits like guild halls, so if a Center is full at any one time you will not be able to join. Trade Centers are rather expensive. I believe they cost the same as a guildhall at comparable sizes, plus an unfortunate amount of papyrus paper. I believe this, the vast improvements that were made with the .info trade tools, and the eventual great success of the large metaguilds led to the lack of use of the Trade Centers already built and the lack of any new Centers. There are two Trade Centers around that I remember: one is located at the House of Bes camp in VoK, the other is near Liberty Bridge. Perhaps you can talk a Bes member into accepting you into their Center if you are interested in checking it out. (According to Ptenkhet, the RPInfo Trade Center is just south of the River Plains U of Worship, and anyone can join.)