The short answer is that when you have an original research manuscript you would like to publish in a peer reviewed journal, the process goes along the following lines:
You choose a journal where you would like to try to publish. The choice is based on the ft between your paper and the journals editorial policy. Take a few that are on the field and see their editorial policy and a few papers they have published, and choose the one that would be a match for your paper in terms of journal scope and level of research. If you're unsure, you may contact the editor and ask for a consultation.
Down load the journal author package and check their requirements for manuscripts and the submission procedure.
Proceed to prepare the manuscript according to the guidelines. Pay attention to formatting, citation and reference style and graphics. The pet peeve of peer reviewers is language, so proof read your manuscript one last time very carefully, or better yet have it proof read by a professional.
When you're ready, follow the submission procedure as specified by the journal. When you submit a paper, it goes first to the editor in chief or associate editor, who checks the paper and if it is acceptable in content and style, forwards it to the reviewers.
The peer reviewers receive the papers from the editor, and give their statement and recommendation whether to accept, with minor or major changes or to reject the paper. The editor takes a decision based on the reviews and sends a decision letter. In most cases the decision is minor or major revisions, if not rewrite and resubmit. If you choose to continue with the paper, then you rewrite the manuscript addressing the reviewers' and editors comments and submit a new version. This goes on usually 1-3 rounds, although you hear stories of even five rounds of revisions, until the editor and reviewers are satisfied with the quality of the manuscript. You can also get a rejection on each round if the reviewers or the editor think you have failed to address the weaknesses sufficiently.
When or if you get an acceptance, the manuscript is forwarded to the copy editors who will edit into the house style and check references one final time. At this time the paper is programmed to a future issue. You also commonly need to check the final pre-print for any errors that may have been done in the copy editing. Some journals also publish the pre-print on-line at this stage.
When that all settled, the paper will be officially published in a future issues of the journal, which may be out in print anywhere between a couple of months to three years depending on the journal's backlog of accepted papers and how many issues they produce per year.
Not to curb anyone's enthusiasm, but rather to set a realistic expectation: All in all, this process takes anywhere between, say, half a year minimum to one year on average and three to five years on the outside for the very demanding very top journals. That is in social science, although your mileage may vary. And all this is after you thought you were ready and done with the paper you have just finished writing.