The name predates the euro by decades and simply meant dollars held in Europe. It now covers dollar balances anywhere outside the US banking system, and the same structure exists for other currencies as euroyen or eurosterling.
It matters to currency traders because it is where the world borrows dollars. Non-US banks and companies fund dollar assets here, and when that funding tightens they must either raise dollars in the spot market or pay up in the fx-swap market, which shows up as a widening cross-currency-basis.
A dollar funding squeeze therefore transmits straight into the exchange rate, which is why the Federal Reserve's central-bank-swap-line arrangements are treated as a market-calming tool rather than an obscure plumbing detail.
Example: a Japanese bank holding dollar assets funds them by swapping yen for dollars. If the basis moves from minus 20 to minus 70 basis points, its funding cost rises by half a percent a year on the whole book.
Related: cross-currency-basis, central-bank-swap-line, reserve-currency, fx-swap