TITLE: How to propose a conjecture
QUESTION [18 upvotes]: What are the basic things (about when and how) to be kept in mind while proposing a conjecture in Mathematics. Should it accompany solid efforts at proof. When should any one think of proposing a conjecture.I'd like to know some of these details just out of curiosity.

REPLY [11 votes]: Conjectures play different roles in different areas of mathematics.  My impression is that they seem to be more common, and perhaps more accepted as a mode of mathematical expression, in highly structured areas such as algebraic number theory, algebraic geometry, and algebraic topology.   My own expertise is in the former two areas, and here there are many very famous open conjectures: the Riemann hypothesis, the Birch---Swinnerton--Dyer conjecture, the Tate conjecture, the Hodge conjecture, Langlands' conjectures, resolution of singularities in characteristic $p$, and so on.  Many of them are named after the particular individuals who first posited them.   Not everyone believes in their validity, but many do, and they gain their legitimacy not only (or not even primarily) from the stature of the name they are attached to, but also from the sense that they fit well with other known and expected results, and with the general directions and themes of research in the field.  Because of the highly structured nature of these fields, it is not so easy to find new statements that fit well with what is already known or suspected to be true!
Of course, the credibility of a conjecture can be enhanced by obtaining evidence in its favour, but some things are difficult to compute, and the credibility of a conjecture often rests on other considerations besides just verification of particular cases.  As I've indicated, a sense of fitting well within the known framework of the subject, while simultaneously extending this framework in a new and interesting way, is often an important criterion for a plausible and interesting conjecture.

As for making conjectures oneself:
Different people view conjectures with differing levels of sacrosanctness,
and this (as well as other considerations) affects their view on the wisdom or otherwise of making conjectures.
For example, in his answer, Theo suggests that it is probably best not to make conjectures at all, primarily for reasons of modesty.   Gerry Myerson suggests that one should at least be proving good theorems in an area before making conjectures in it, which is certainly sensible advice.  
My own view is the following: I think that it is good to make conjectures if you sincerely believe them; they help to focus (one's, and other people's) research, and draw attention to what is important in a given line of investigation.   However, just in the interests of not damaging one's own reputation too much, it is probably not sensible to make a conjecture unless one has a very good feel for the subject in question (meaning that one should be proving theorems in the subject that are near to the cutting edge).  Even when one is at the cutting edge of research, there is always a possibility that one's conjecture will be wrong, due (typically) to a misunderstanding of or lack of knowledge of certain phenomena and (counter-)examples.  (This is an occupational hazard of making conjectures!)
If one suspects that a certain phenomenon might be true, but is unconfident in one's intuition, or unsure whether it is always true or what the precise hypotheses might be, one can always raise a question instead, rather than stating a definitive conjecture.