The Trade Window is opened whenever you click the "Speak To" under the picture of the leader of another race or when clicking war related events that appear on the event stack. You must have researched the Universal Translator and have made first contact with a race in order to trade with it.
Using this window, you may negotiate treaties and trade resources or technologies. Tradable goods include money, influence points, Trade Goods, technology, planets, starbases, ships, or treaties. Treaties include peace treaties, alliances, or offering to declare war or make peace with another civilization.
The AI has a value attached to each possible trade item. High diplomacy and good relations with the other race will give you an advantage in trading, and thus civilizations you have joined in an alliance with will often agree to obviously unbalanced trades in your favor simply because they like you. Watch the color of the response text when forming a transaction, as the color is red while the AI will not agree to the deal and green when it will agree to the current transaction. Use the color coding to see if you can gain a bit extra to an already approved deal.
The technologies listed are only those which one side has that the other doesn't, and thus the trade window can be used to discover what technologies any other race has and likewise where you hold a technological advantage.