How involved was Einstein in making the nuclear bomb?

He co-wrote/signed a letter to Franklin Roosevelt in August 1939 suggesting that the government look into whether nuclear weapons might be something to worry about. It neither advocated making nor using the atomic bomb; it was very modest in retrospect. It led to the formation of a committee in the National Bureau of Standards, the Uranium Committee, which coordinated some research in the US into the matter. It didn't go anywhere particularly fast. Later, for different reasons, the US work got taken over by different people and this later effort eventually morphed into the Manhattan Project to make the atomic bomb. Einstein was not involved in this later effort; he was considered as someone they might bring in, but they chose not to for a variety of reasons (his politics were suspicious, and his physics is not particularly useful for the practical job of making nuclear weapons).

Which is to say: Einstein played a role at the very beginning of the process. It goes too far to say he really "started" the Manhattan Project, though; the Uranium Committee was not the Manhattan Project, and arguably it actually slowed down the effort. The actual Manhattan Project's impetus came from a different direction (the British MAUD Report).

I have written a bit about the irrelevance of Einstein's physics to the atomic bomb, here — it is a common misconception that his scientific work (e.g. E=mc^2 ) was vital to the work. It was not (atomic bombs are non-relativistic phenomena, and E=mc^2 just tells you why they don't violate the conservation of energy and mass, it doesn't tell you whether bombs are possible or even how they really work).

You might followup with the question of: so why do so many people think he was crucial or important to the effort? And therein lies a more interesting cultural question: what work is done by putting Einstein so central? For one thing, Einstein was already world-famous, and his popular image plays into many myths about lone geniuses. For another thing, the connection between the seemingly abstract and non-practical theory of relativity and the very concrete and world-changing atomic bomb makes a nice narrative about the value of basic science. For another thing, if Einstein thought it was a good idea, then who can disagree? So the government officials enjoyed pointing to him as well for this reason, because it made their later decisions seem unquestionable. But the reality is, as in most things, more complicated.