Entering students often think of a PhD as a 'magnum opus', a brilliant research project culminating in a great work. This is rather a demanding model and few students win Nobel Prizes as a result of their doctoral studies. As one colleague phrased it, a PhD is less like hacking through the jungle with a machete, and more like crawling around on the ground with a magnifying glass - less major discovery of new lands, more painstakingly detailed investigation of familiar ones.
More realistically, a PhD is a demonstration of research competence. There are certain things that you are demonstrating through your dissertation:
These reflect competence and professionalism, rather than greatness. Importantly, they are as much about comprehending others' work as about doing one’s own.
So, a PhD is research training leading to a professional research qualification. A PhD can be a deep, specific education in a discipline, preceding a postdoctoral period of on-the-job training (note: there is no implication that the PhD is the end of your education or training). It involves you doing a substantial chunk of research, writing it up and then discussing it with professional academics. You have a supervisor (or two, or more) to help and advise you, but in theory at least the PhD is something for which you take the initiative, and so it is a demonstration of your ability to do proper research independently. The process is rarely smooth; along the way you are likely to learn a great deal about how not to do research as well as about how to do it effectively.
At a sordidly practical level, the PhD suggests that you are good enough at research to be appointable to a university post. A PhD is highly advisable for a career as an academic, or helpful for a career as a researcher in industry. PhDs are recognized around the world and tend to have pretty good quality control, MD a PhD from one country will be recognized in another without too much snobbery. Still at the practical level, if you have a PhD, you usually go onto a higher pay scale.
There are other views of a PhD, as well. It can be viewed as an initiation rite, in which you undergo an ordeal and, if you come through the ordeal in a creditable manner, are admitted to membership of the academic clan. Continuing the analogy, having a PhD will not be enough to make you a Clan elder, but it will mark the transition to full adulthood. You are treated differently if you have a PhD - there is a distinct feeling of having become 'one of us'.
The 'rite of passage' is not just a snobbery thing; the ordeal (and the education) give you a different way of thinking about things. A PhD can be viewed as one's entry into the research discourse (which equates roughly to the research community's dialogue about what it believes it knows and has a good basis for knowing). What it should do is prepare you to consider and debate what you know and how you know it. This means that you'll have developed your critical thinking, that you'll have learned about weighing evidence and questioning assumptions. You will gradually notice a different way of thinking about things - for example, when you start making administrative decisions in your subsequent career. A good example of this is undergraduate student projects: in many departments, staff with PhDs typically want to use the projects as a way of teaching the students how to conduct research, and staff without PhDs typically want to use the projects as a chance to give the students an industrial placement. The PhDs' view is that the students need to learn critical thinking skills valuable for later life; the other view is that the students need to be equipped to find jobs. Which is right? This is a good question, and one which would take us on a lengthy diversion. The main point is that doing a PhD does change you.
So, a PhD can be many things: research training, springboard for specialist expertise, rite of passage, job credential ... what it means for you depends on which opportunities you seize, whether you keep an eye on `the Big Picture', what sorts of relationships you form and so on.
There are various types of research degree; what they have in common is that they involve research by the student as a core component. This is different from a taught degree where there may be a research project (for instance, an MSc project), but where this research project is only one component among many on the course.
Strictly speaking, a research degree involves a thesis, which is the argument that you propose as a result of your research. Again strictly speaking, the dissertation is the written document which describes your thesis. In common usage, the dissertation is often referred to as 'the thesis'. It's worth knowing about -the distinction in case you have a particularly pedantic external examiner - it helps you get off to a better start.