Which institutions are in the eurodollar market?
Eurodollars are, by definition, held (on the liability side) inside the banking system. They're US dollar deposits held either outside the US or inside the US via an International Banking Facility. On the asset side, eurodollars are typically lent by nonbank financial institutions. This means that the eurodollar market links the two segments of financial markets: it's typically nonbanks providing short-term funding to banks.

For an introduction to nonbank financial intermediation, I'd recommend "Shadow Banking" (Pozsar, et. al., 2010).