Why do people publish on arXiv instead of other places? I work in engineering and I was warned by a colleague years ago that I should not cite materials on arXiv because they are not peer reviewed. Although it was never clear to me what peer reviewed meant because the papers on there seemed to be well maintained and the ones I have came across has been theoretically sound. Academia stackexchange certainly didn't help in relieving some of those anxieties with people asking if they are "ruined" for posting on arXiv. 

One can understand my point of view better by considering that in engineering there are well established conferences and list of papers published through those conferences (although not all high quality). Browsing through a list of citations published by engineers, there is a definitive lack of "arXiv" related papers. This is exacerbated by the fact that arXiv papers seem to be less applied and more theoretical, "mathy", so they are largely hidden from view for the typical engineer. 

But from my recent experiences, it seems that a few fairly notable researchers in my field have extensive track record of publishing materials on arXiv. People such as Steven Strogatz who works in nonlinear dynamical systems and John Baez. I have hypothesized in the past that very well established researchers are submitting papers on arXiv instead of science journals because they would like a wider audience, and undergraduate researchers are submitting on arXiv because there is no other venue for them. But of course I am not sure if that is indeed the case. 

Now I am currently reading some material published on arXiv by someone who I am not very familiar with. I am still not 100% confident on the "quality" of the paper i.e. scientific accuracy of theoretical and experimental results. But I believe that I would be more open to using materials on arXiv in the future if I can understand the motivation behind submitting to arXiv i.e. whether if they are just low quality papers without any other place that accepts them.  

Can someone please offer a break down of some the reasons why one would publish to arXiv?
The major use case of arXiv is for disseminating manuscripts that you also publish in a journal or conference. By posting a preprint on arXiv, people can find your research, build on it, cite it, and give you feedback on it immediately, while at the same time the same work goes through the (sometimes slow) peer review process. Some of these papers will fall out of the peer review pipeline at some point, and only appear on arXiv, but that doesn't necessarily mean that  they are less useful, important, or sound.

To give a specific example, in the last year or so I have been working in an area so new that most of the relevant research is still only available on arXiv.  I expect these papers will eventually appear in journals too, but the slow peer review process means that the latest journal issues do not represent the state of the art for this particular topic. Conferences have a quicker peer review cycle and are more current, but most only publish relatively short papers.

arXiv is also useful for work that is in a format not suited for a conference or journal (e.g. a thesis), or for extended versions of papers that are published somewhere else.

I would caution against rules like "I should not cite materials on arXiv because they are not peer reviewed." Peer review does not guarantee sound, high quality research, nor is the inverse true.  You should critically evaluate each paper, peer reviewed or not, on its individual merits. (Also see this related question on Math Overflow.)