There is no one definition of the exact point at which the Western Roman Empire collapsed, but a solid contender is the sack of Rome in 410. This seriously hampered the already tenuous ability of the city of Rome to project influence and power into more distant lands, as it had for hundreds of years previously.

Of course, there never stopped being Romans. The city has been continuously inhabited until today. The political identity shifted from empire into a city state closely tied with the Catholic church, into a regional power (the Papal States), and into the capital of Italy as it became a nation.

Roman language (Latin) never disappeared, as it was the liturgical language of the Church, and it never stopped being used and studied by educated people throughout Europe. Roman culture, which was largely 'civilized culture' for most of western Europe, was partially assimilated and mixed with local cultures as they developed into their own independent political entities. The Byzantines never stopped considering themselves Romans, although their culture had been solidly Greek for several centuries already.

As for how long people continued to consider themselves Roman in other parts of the world? The idea of being Roman was very attractive to people, from Spain to Anatolia. For example, Charlemagne wanted to be crowned Emperor of the Romans, despite being a Frank and only recently having conquered the city of Rome. Why? Because Rome was considered such a prestigious and desirable title. Hell, even 1000 years after the Roman Empire was considered collapsed, Mehmed II took the title of 'King of the Romans' after conquering Constantinople. So its safe to say people continued to consider themselves Romans for a long, long time after, although what exactly that meant evolved with time.