The Research Lab allows you to research new Technologies. Each level of the Research Lab allows faster researching times and (up to level 12) opens up research into additional new technologies.
Having research labs on multiple colonies does not allow multiple research projects to be conducted at the same time: you are limited to researching one new technology at a time throughout your entire empire. However, once a technology is researched, it’s benefits apply to all of your planets immediately. Put another way, the knowledge gained through research is shared by all the planets in your empire. With the Intergalactic Research Network, research labs on several planets may be joined together so that they can research new technologies much faster (but still limited to a single research project at a time).
Note: While a Research Lab is under construction, no research can be conducted. If a research project is underway, the research lab cannot be upgraded until the research project is complete.
The building costs of the research lab doubles with each new level, though costs for each technology researched will vary. The resources needed to fund a research project are deducted from the planet when the project is started. To see what Research Lab levels are required for a given research project, see the Technology Tree.