I am a neophyte amateur mathematician. I have been reading a lot about journals and the topic of peer-review in mathematics journals. Does one have to have professional credentials or have a Doctorate in order to publish in peer-reviewed mathematics journals or just the desire to compently solve mathematical problems?
[Math] Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal
publishingsoft-question
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http://plus.maths.org/ describes itself as "an internet magazine which aims to introduce readers to the beauty and the practical applications of mathematics." It's very well written. The level of the maths varies, but I think you'll find some content that's at the right level for you.
And to echo @Alon Amit's comment - Martin Gardner was a very lucid and entertaining maths writer, and several of his books and magazine columns are knocking around on the web in one form or another.
This really should be a comment: but it is a bit long and perhaps important enough (since it addresses a common misconception) so I will post it as a community wiki. I want to address the mention of Perelman's proof in the original post.
Firstly, one should be aware that Perelman, despite his current somewhat indistinct status, is and was a professional mathematician. He has a pretty impeccable academic pedigree from Soviet Union, and has worked in research positions in major North American universities in the 1980s and 1990s. Furthermore, the geometrization conjecture is not his only contribution to the field: in 1994 he was already quite renowned for having proven the Cheeger-Gromoll Soul Conjecture which stood open for 20 years, and he was also known for his work in comparison theorems in Riemannian geometry. So not only was he a known variable to the other professional mathematicians on a sociological level (having worked both in the east and in the west [using the cold-war era splitting]), he was a well-known variable in the sense that he has already demonstrated extreme technical proficiency. (Note, his proof of the soul conjecture was published in traditional manner in the prestigious Journal of Differential Geometry.)
In short: Perelman is not just some nobody who makes an astonishing claim.
Secondly, it was not clear from the get-go that Perelman did not intend to publish his proof in traditional media. Furthermore, he actually did not ostensibly claim a proof of the geometrization conjecture (emphasis mine):
... We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
(Note that the above e-print was followed by this and this.) Furthermore, one should factor in the fact that for the prestigious Annals of Mathematics (where results of this nature would usually be submitted to), the average time from submission to acceptance is 19 months, with the average time from submission to print being 3 years. Many of the groundbreaking results in mathematics would've been read, torn apart, verified, understood, and possibly even improved upon before the paper copy actually hit the shelves (in this case it took a little bit longer, the full verification was completed by several groups working independently by 2006; this is partly due to one particular theorem in Perelman's papers that was only accompanied by a particularly sketchy proof). That is to say, whether a paper is important is usually not judged by the fact that it appeared first as a pre-print on arXiv.
Lastly, a very important point that is often glossed over in the popular account of such stories (not just that of Grisha Perelman but also that of Andrew Wiles) is that despite the solitary working styles of the main protagonists, they do not conjure their solution out of thin air entirely by themselves. In particular, based on already known works there already exist compelling evidence that the pursued line of attack has some chance of working. In the case of Wiles, he did not prove the FLT per se: instead he proved the modularity theorem for semistable elliptic curves. Following a strategy described by Gerhard Frey in 1984, two ingredients were needed to settle Fermat's Last Theorem. The first ingredient was concretely identified by JP Serre and was called the epsilon conjecture (now Ribet's Theorem) and was demonstrated to be true by Ken Ribet in 1986. The second ingredient is what is now called the modularity theorem, which was conjectured in the 50s and 60s as a general statement about elliptic curves (only some special cases of the general statement is needed for FLT) before its close connection with Fermat's Last Theorem was recognized. So in Wiles's case, the approach is already previously well justified, and that the technical ingredient ought to be true is already expected/hoped-for based on other previous works in algebraic number theory.
The setting for Perelman's work is not all that different. That Poincare conjecture is a subcase of Thurston's geometrization conjecture is well-known (and trivial). That geometrization holds in certain special cases (hyperbolization for Haken manifolds) was proven by Thurston. And as mentioned in the quoted abstract above, Richard Hamilton developed the Ricci flow at least in part due to his interest in the geometrization conjecture, and has established a program and identified the main technical ingredient needed (a good procedure for Ricci-flow with surgery) to use the Ricci flow machinery to prove geometrization. So at the level of the "large picture" the approach seems reasonable. Furthermore, the main insight, the "entropy formula", gives a nice and tangible description of the mechanism which drives the proof. While the details still needs to be checked, it is something that looks believable, especially given some of the justifications derived from theoretical physics.
Best Answer
No, no sort of degree or specific credentials are required. It is far from unheard of that someone publishes one or more papers while still an undergraduate. There are even a few instances of important work being published by high school students.
If you are an uncredentialled mathematician you should take extra pains to make sure that your work passes superficial tests for looking serious and competent. Off the top of my head, this would include the following:
Make sure that your grammar and spelling is virtually flawless (in particular, you have some spelling mistakes in your post, which is probably not such a big deal for this site, but would create a bad impression in a journal submission).
Use some form of TeX to format your document. (This is freely available online and just takes a little practice to become reasonably proficient. Not to do so is a big warning sign of outsider status.)[1]
Make sure that your work contains a bibliography which gives due acknowledgment to previous work in the field. Wherever possible, you should try to cite primary sources (i.e., journal papers) or standard texts, and you should try to avoid citing things like wikipedia and popular books.
[1] I am aware that there are a few leading mathematicians who do not use TeX. The geometer Cliff Taubes is a popular example. However, Professor Taubes began his mathematical career before TeX became widespread and, in any event, does not need any help in getting his work taken seriously. I am not saying that someone who does not follow the above precepts is not a serious mathematician, just that they look less like one at a first impression.