Since I come from a small department that barely has any experimental research going on, and no research in my areas of interest, I looked outside my school and on the recommendations of my professors, got a paid job doing research at my local NASA branch. It is all directed by a senior research scientist here, but he does not do any lab work. Our group consists of 1 post-doc, 4 graduate students, and myself.
Our boss (the senior research scientist) directs the research, although we meet regularly and discuss ideas and in this way, the direction the research goes is very much influenced by the students. If we have a good idea that we have the resources to investigate, we go for it. It is our group of students that design the experiments, we order the equipment we need, we assemble it, we perform the experiments, we collect the data, we interpret the data. For each project, our boss assigns one or two of us to direct the research, and on its completion, that student writes the paper and becomes the first author.
Essentially, our boss is responsible for organizing our group, deciding our primary areas of research, assigning principle investigators for each project, keeping us directed and on a time frame, getting funding and keeping up with the competing research in our area, providing reports of our progress to his superiors, dealing with the mess of bureaucracy so we don't have to, and helping us edit and submit our publications.
When a paper is published, the authors it lists are typically all of the contributing scientists, ordered with respect to their respective contributions. When I say "first author," I mean the first name in that list of authors, indicating that you were the *primary* writer of the paper's content and usually, the primary researcher of the project.
For the paper on which I am listed first on the list of authors, my boss and I discussed what we wanted to do, another student in our group fabricated the samples I needed at my request, I determined which samples were tested, I determined what tests were run, I ran the tests, I collected the data, I analyzed the data, I wrote the paper, my boss helped me edit it, and then we submitted it.
You do not have to be working independently on your own ideas to be doing research. If this is what you are doing, then you are doing *independent* research, which is impressive, but not all research has to be independent. When most people refer to doing undergraduate research, they refer to joining a research group, usually directed by a professor, and assisting with that research. In some instances, this may mean just collecting data. Usually, if someone is the first author on a publication, it means they did whole lot more than collect data. I think in most instances, research is performed in groups consisting of a professor, perhaps collaborating professors or collaborating professionals, and several students.