Technologies represent new scientific advancements or inventions for your civilization. Each technology you research unlocks new buildings, units, or other advancements, and expands the power of your nation.
Technologies are developed through a continuous research process, the key to which is the Science stat. However, they can also be "hurried" through Eurekas. For more info on that, check the Science article.
The Tech Tree
Similarly to previous games in the series, technologies are organized into a "tech tree," which means that certain technologies are prerequisites for others. Technologies which are available for research appear dark gray, while the ones not yet available appear light gray. Technologies which have been researched already appear teal. In most cases, all technologies form a continuous system, where each one is connected to others, both lower down and higher up the tree. Unlike previous games, you can now advance continuously in some areas, while leaving behind other areas (i.e. not all areas of the tree are connected to each other, and are thus not required for advancement). However, if you're striving for a Scientific Victory, you will have to develop the entire tree.
Also, there are a few technologies that do not lead further up the tree, which are known as "leaf techs." You can disregard these techs if you have no particular use for them - for example, Celestial Navigation is a leaf tech, and if you have a landlocked civilization you may not need it at all.
Unlike previous games, the technological tree is not the sole means of advancement for a civilization! There are now two separate development trees: a "technology tree," focused only on the civilization's technological development, for things like tools, techniques, machines and weapons; and a "civic tree," focused on a civilization's social and political development. Thus, a civilization has to advance in two separate trees at the same time.
Era progress bar
Underneath the Tech tree itself the scroll area also contains the Tech progress bar, which depicts the overall progress of the game (with the current turn marked on the bar), and where approximately each civilization stands in terms of tech development. You will see a number of circles on the bar, each one marking one or more civilizations. A circle marking a single civ bears its leader's portrait, and one marking multiple civs a number; scroll over each circle to see the exact Tech Era it shows, and (if it shows multiple civs) which leaders are currently in it. This way, you can have a good idea of where each civ stands in terms of development compared to the other civs.
The base cost of each technology (and its cost modified by the current game speed) is displayed in-game. It ranges from 25 Science for the first technologies to 2600 Science for the most advanced ones.