Terms like 'Iron Age', 'Bronze Age', and 'Neolithic' are common in popular accounts of the history of humankind. The idea of different 'ages' of humankind is a very old one; Hesiod, the 8th century BCE Boeotian poet, used it in Theogony, and the idea of a 'Golden Age' is actually a direct inheritance from him. It's also an idea found in Popol Wuj*, the K'iche' Maya creation myth, in which the gods created different generations of humans from mud and wood before settling on the final form. But in archaeology, the ages we're all familiar with go back to the first rigorous formulation of the discipline in early nineteenth century Denmark, when items found in graves were placed in a rough chronology based on the dominant materials used in their creation.
It was an early form of seriation. This involves dating items in a series on the basis of their immediately-obvious properties, like size, shape, and style, and their presence in conjunction with other material at the site. The archaeologist can then attempt to tie this relative chronology to absolute dates using written accounts or other reasonable dating principles where possible. The first part provides a relative chronology - when certain items and styles were in use in relation to one another - and the second is supposed to provide an absolute chronology, marking the absolute position in time in which the items were used and the lives were lived.
Let's say you found a bronze spearhead in a grave in Romania, and you know that identical spearheads have been found in sites in Greece with an approximate date of 1200 BCE. You may then infer that the spearhead dates to that time and may be an import from Greece; you might be mistaken, but it gives you a reasonable guide to work with. You can then look for other graves with similar items in Romania, and you can also look at the other items present in the original Romanian grave to see which other things were probably being used in the area at 1200 BCE. This allows you to approximately date, or at least serialise, other sites in the area by the presence of items similar to those known to have existed around 1200 BCE. That's the theory, at least; in practice, separating out different eras of occupation or use of a site can be tricky.
It's an approximate and not entirely accurate method of dating, and various chemical methods are now employed. Radiocarbon dating is the most famous, and it provides relatively accurate dates for most testable items within the last 40,000 years or so. Seriation isn't unimportant by any means, and it's always good to have many strands of evidence in making a chain of argument about the past, but the crude series 'Neolithic' --> 'Eneolithic'/'Chalcolithic'/'Copper Age' --> 'Bronze Age' --> 'Iron Age' is certainly no longer very useful. One of the reasons it is used in popular accounts, and even some more academic works, is that it provides a comprehensible structure to the unfamiliar events of prehistory, and to that extent it can be a reasonable way of providing information about the past.
But it simply isn't true that technologies turn over so quickly and so totally. Use of bronze doesn't and shouldn't define a group of people and their activities. What is true of one bronze-using group is not necessarily true of others, especially as different groups of people can learn from one another. People who one generation ago used bamboo blades to cut pigs open and stone celts to cut down trees might today use their iPads to record nature videos.
Nor can we infer that all bronze-users lived at the same time. Different groups of people began using bronze at different times, so 'Neolithic' is not an absolute period of time. It applies in only one place at a time. And there are, of course, people using stone tools for most major activities even today, after other humans have flown into space on rockets. In the end, the use of bronze tools carries only one necessary implication and none others: that the group possessed bronze, and possibly the technology to produce it. It doesn't mean anything else. And it is inherently spurious to base a chronology on the occasional use of one technology.
The boundaries between these ages are usually quite fuzzy. The 'Neolithic' is sometimes thought to have begun when agriculture began, or when a particular style of stone celt was used. Sometimes it refers to human groups who used ceramics. It is sometimes thought to correlate with innovations in social structure or a growing complexity of life, but there isn't a necessary correlation between changes in axe technology or the adoption of ceramics and changes in the complexity of social structure. In South America, many of the earliest sites demonstrating large populations, public architecture, and complex social structure are pre-ceramic and non-agricultural; they depended not on a surplus of farmed goods, but rather on the fruits of the sea, sealife being particularly abundant on the Pacific coast of Peru, and they didn't use pottery to cook or store their food, as they did not possess it. Were they 'less advanced' than bronze-using tribal pastoralists? That doesn't even seem like a valid question to me.
The first users of copper and bronze tools did not use those materials for everything - they didn't suddenly chuck away all of their stone tools or the knowledge of their manufacture. Almost uniformly, they used knapped and ground stone tools alongside those made of bronze, or reserved the use of bronze only for special items and occasions. This is a pattern found in China, when as late as the Zhou dynasty (c. eleventh century BCE) ordinary people still employed stone tools for most ordinary tasks. It was true of the steppe pastoralists associated with the spread of Indo-European languages. It was also true of people throughout the Americas who had, by the time of European contact, acquired knowledge of some form of bronze. They all used stone, copper, gold, and anything else they found useful, just as people do today.
It's also important to recognise that many groups who have been classified as 'Stone Age' used a diversity of materials for their tools, with stone being only one. Shell, bamboo, wood, tooth, bone, feather, bark: more materials are available to the metal-less human than merely varieties of solid aggregated mineral. Incidentally, the 'Stone Age' view was once taken to such extremes by prehistorians in the nineteenth and early twentieth centuries that certain human groups in Polynesia were believed to have 'regressed' from using stone to using shell. They were called 'Shell Age' peoples wherever a lack of stone on an island caused the population to resort to the use of much more abundant shell. This technological innovation in the face of resource-scarcity became a simple case of socio-cultural atavism in the minds of certain prehistorians burdened with their ridiculous views on the nature of cultural change.
Such absurdities are now uncommon in academia (the so-called evolutionism behind it all lost credibility a long time ago), but the idea of different ages of humankind continues to find employment in popular anthropology. I think that's probably okay. But it's very important to realise that these are not different ages or different worlds defined by the widespread use of a single, powerful, overriding technology. They're merely occasionally-useful attempts to create a relative chronology based on the growing use of certain materials. They don't necessarily correlate with innovations in art, society, or other aspects of culture, and earlier 'ages' certainly don't imply barbarity, extreme patriarchy, or lesser intelligence.
I've recently been trying to get the chronology of humankind straight. There are very few easy points of reference. The terms 'Neolithic', 'Chalcolithic', 'Space Age', and others like them, definitely do not fit the bill. Cultural change is normally continuous. It simply doesn't break up easily into neat segments, however useful the idea was in giving some shape to prehistory for those lacking modern dating methods.
* 'Popol Vuh'; the 'j' is a Spanish j, (IPA: /x/).