This describes any international corporate or government bond that is denominated in a ‘eurocurrency’. Confusingly, given the name, this in turn is any currency (not just the euro) held outside its country of origin.
So, while most dollars are held in the US, ‘eurodollars’ can be kept on deposit across the world; the same goes for any major global currency. This means a US company seeking to borrow dollars cheaply can do so by issuing eurobonds in different regions, such as Europe or Asia; it is not limited solely to using the US bond markets.
The exact origin of the term eurobond is debated, but it is thought to come from a post-war deal that involved the Russians issuing dollar-denominated bonds in Europe, rather than America.