Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal? 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?  
 A: There have already been a lot of good advices. Let me stress the importance of chosing the journal right. There are too difficulties here:


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*The journal is interested in your subject.

*Your paper fits the quality standards of the journal.


As for (1) there is an easy algorithm: you sure cite other works in you bibliography, right? Look at the journals in which they were published. If an specific journal does appear more often than others, there is a good chance that your subject interests that journal.
The second issue is more difficult to grasp for beginners without the advise of a senior colleague. It takes some time to develop the sense of how good is your paper and, consulting the JCR list of mathematical journals, find the appropriate journal according to its quality. Of course, you could try another easy "algorithm" for this: start with the highest quality journal in your field and, as they reject your paper, go down the list. There is nothing wrong in doing this, but some experience could help you in not trying Annals of Maths as your first option.
And a final advice: I'm sure that almost every professional mathematician has had at least one of his/her papers rejected (in my case, I think that 30% of my papers were rejected by the first journal where I've sent them).
A: 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:


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*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.  
A: There is usually no technical requirement for any particular credentials, and it seems unlikely to me that editors and referees will check for this. The usual publication process is that you submit a paper to a journal editor, who then forwards it one or more competent referees, who make a recommendations on the suitability for publication. Often, the referees make suggestions for improving the paper, whether or not it is found acceptable for publication.
Nevertheless, an amateur mathematician will find him or herself at a disadvantage in several respects of this process. First, for someone with less experience in the professional mathematical community, it may be more difficult to recognize which accomplishments actually merit publication, and editors sometimes receive articles submitted in good faith from amateurs, which are trivial or seriously wrong in some respect. Second, even when the result is correct and worthwhile, if the presentation of the paper deviates from the accepted norms, it may be found wanting. For example, it is more difficult for an amateur to know which topics need more careful explaining in a paper and which do not, and inappropriate decisions in such cases can hurt the reception of the paper. 
I see (as I write this) that others have now given some concrete advice. The most important advice I can give is to take the suggestions of the editors and referees seriously. If someone claimes that part of your paper is unclear or wrong, then you should try to understand exactly why they thought so; doing so will inevitably lead you to a better paper.
A: In the math department at MIT, there's an annual award for best published paper by an undergraduate majoring in math.  Most undergraduates lack credentials other than a high-school diploma, and one must suspect that most undergraduates majoring in math, if they have professional credentials, are credentialed in other fields.
A: I've never published a paper, so I'm the wrong person to answer your question; this answer is really more of a long comment.
A few years ago, I had worked on something independently that I thought I'd try to get published. Looking back, it wasn't very interesting--basically an attempt to generalize Riemann integration to an abstract setting---but I tried submitting it to the Proceedings of the AMS. It was rejected, but I got about two pages of helpful and interesting comments from a referee (suggesting that I try a non-researchish journal; I never actually had the time to do that though) as well as kind words of advice from the person sending the email. So it was a good experience in the long run. I also got practice in writing concise mathematics and with LaTeX (though I was fairly fluent by that point).
Anyway, the point is that even if it gets rejected (which it won't necessarily), you'll probably learn something about mathematical life from the experience. And, besides, you can put it on the arXiv if someone "endorses" you.
A: No professional credentials or degree is required. However, lacking such will subject your paper to much higher scrutiny so it is essential to adequately prepare for such. Since many uncredentialed submissions are by cranks it is important to differentiate your paper from such submissions. 
First, be sure to choose an appropriate journal. Browse the recent publications of various journals to see if they accept papers on your subject at the level of your work. Seek advice from professors at local universities or in online math forums. 
Second, make it clear that you are aware of prior work by properly referencing such. Consult mathscinet and Zentralblatt as well as web searches (e.g. Google Scholar) to find related prior work. You may be able to access mathscinet at a local university library.
Crank submissions usually fail on both accounts. Namely, they often submit to journals whose prestige far exceeds their results and they usually fail to adequately cite prior work. So if you succeed on both these points you're off to a good start differentiating your work from a crank submission.
Best of luck!
A: Implicit in the previous answers: do your best to pick an appropriate journal. Even professionals don't always know whether a particular journal will publish their paper (even if the results are perfectly correct). There are too many factors to count that can affect the editors' decision. For an amateur, the choice can be even more opaque. 
Graduate students often rely on advisors to guide them in picking the right journals for their first papers. If you have academic contacts, they may be willing to give you advice about possible journals. 
There has been a heavy emphasis on undergraduate mathematics research recently, at least in the United States. Several journals have sprung up to publish papers written by students. If this applies to you, search for "undergraduate mathematics journal" on Google to find several possibilities. 
A: Pete Clark has given an excellent answer. I just want to add two more things:


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*Try to discuss with trustworthy professional mathematicians before publication who will be sympathetic and honorable.

*First publish in a website like the arXiv. It is free-for-all, and you will get to know in short order if your paper is really interesting.
A: No, but reputation counts for a lot and not having an academic track record means that you are missing many of the easy credibility markers.
Interestingly, when I was working towards my PhD back in the mid 1990s my supervisor made an interesting observation that some of his best work in fact had been the hardest to get published.  He felt that really innovative, ground breaking ideas were often the most difficult to get accepted and were subjected to far greater scrutiny than normal.
