Orphan Black has several key Organisations/Projects with various code names. I find myself getting muddled between them. Particularly after a long gap between the seasons.
I would like a summary (in a sentence or two) of what the following are:
All of the things you mention are different aspects of the same basic "project" -- human cloning. By the time of the current events in the series, it has been splintered into a number of different pieces.
Originally, there was a single genetics research project investigating the idea of human cloning. This original experiment was a military-funded operation, which was not unusual at the time. By military, I assume they are talking about DARPA - the US military research division (the same one that invented the Internet, for example.)
At some point in the past, this project was supposedly de-funded and shut down. In reality, it was taken "underground" and split off into two distinct branches:
Project LEDA, operated by the privately-funded Dyad institute, was focused on creating clones of a female human subject. Their one success leads to the main characters in the Series; in addition to trying to replicate that success, they continue to monitor the health of the existing batch of clones, as well as investigate things that went wrong (e.g. the fact that one of them is not sterile, the fact that they are at high risk for fatal lung disease, etc.)
Project CASTOR continued to be operated by the military, and was responsible for trying to clone a male human subject. Prior to Season 3 we know almost nothing beyond the existence of this project, and a couple of the clones.
Topside is a fairly unknown quantity at this point, but they are effectively the private corporation that owns and operates the Dyad, or at least, they have a significant amount of control over how the institute runs.
The names Leda and Castor are both pulled from ancient mythology: Leda was the name of a woman who was impregnated by Zeus, who had taken the form of a swan. Castor was one of the four children (two sets of twins) that hatched from the resulting eggs.
