Nationals of most European countries do not need a visa to enter the USA but must apply for permission to travel to the USA - ESTA. Once they are approved via ESTA they can enter unless turned away on arrival. This would never be due to not having a visa but for some other reason.
Those nationalities requiring a visa cannot travel without it either because the ESTA process is incorporated in the "visa" process.
ESTA is in fact the same as a visa, no matter what it is called, although perhaps a lower level one in terms of ease of application - except that by moving the cordon (and writ of the USA government) out to foreign airports the concept behind ESTA actually increases restrictions on foreign nationals. It is a reintroduction of travel restrictions on those formerly at liberty to travel and enter without a visa. A "visa" or waiver of its requirement is an official approval to travel and enter, not having the ESTA clearance is equally restricting as not having a "visa".
International protocol dictates that visa requirements are reciprocal and lifting the requirements is normally reciprocal. A visa requirement is the default position, doing away with it is a case of bi-lateral agreement and leads to free travel. Introducing a visa requirement normally invokes the other country also stipulating a visa requirement. ESTA is so called on the basis/pretext of national security (like PRISM and all the rest) and it is clearly a blanket travel restriction. It is likely deliberately not called a visa so as not to provoke reciprocal visa considerations and/or in order to avoid evoking parallels with its nemesis, the USSR which required visas of everyone except its very closest allies. But ESTA's effect is precisely that of a visa - everyone has to get one.
None of this should be confused with permission to reside and take up employment, called a green card in the USA. That is an altogether different thing. When the USA lifted visa restrictions on most European nationals they at the same time permitted a limited length of stay, but not to take up employment. With ESTA, the same still applies. Whether you have ESTA or a visa, you can still be turned away at immigration - not a high likelihood, but a possibility all the same.