Contrary to the name, Eurodollars have very little to do with the Euro or European countries. Eurodollars are U.S. dollar-denominated deposits at banks outside of the United States. This market evolved in Europe (specifically London), hence the name, but Eurodollars can be held anywhere outside the United States.
The Eurodollar market is relatively free of regulation, and so banks can operate on narrower margins than their counterparts in the United States. As a result, the Eurodollar market has expanded largely as a way of circumventing regulatory costs.
The average Eurodollar deposit is very large (in the millions) and has a maturity of less than 6 months. A variation on the Eurodollar time deposit is the Eurodollar certificate of deposit (CD). A Eurodollar CD is basically the same as a domestic CD, except that it's the liability of a non-U.S. bank, and they are typically less liquid and so offer higher yields.
The Eurodollar market is obviously out of reach for all but the largest institutions. The only way for individuals to invest in this market is indirectly through a money market fund.