Your question is actually two. How are articles typeset and what can be used to type set articles.
To answer the second, how you can accomplish this, there are three basic choices (1) a word processor such as Word or , for example the word processor in Openoffice, (2) (La)TeX, both suggested by others and (3) a typesetting software such as InDesign, Quark Xpress, Scribus or Publisher. Out of these choice (1) is the weakest and (2) and (3) provides much better control on the final product. There are freeware options in all three categories with LaTeX being the only free core. Option (3) is constructed for very flexible design and is used throughout the graphics industry for typesetting. LaTeX has huge advantages to automate advanced type setting and also handle both long complicated documents as well as documents with common elements such as scientific articles. Option (1) is perhaps the easiest to start using but is not intended for advanced typesetting such as for example in academia.
To the first question, how are research papers typeset? Many, if not most, journal publishers use LaTeX to set articles. In fields where equations and specific scientific notation is frequent, it will be used almost exclusively. I doubt many prominent journals would use word to typeset their final product but many probably use different forms of specific typesetting software.
The choice of typesetting method is, however, not necessarily coupled to how the author needs to format their manuscripts. A journal typesetting in LaTeX can easily accept word files. Remember that a manuscript is rarely provided in the final layout of the journal, the layout is something the journal does in the end. I know journals have provided cuts in pricing for those who provide type set articles after review but for manuscripts a single column wider line spacing text is all that is required. This means that the choice of format for the manuscript is less important although, if a journal provides a template or class file for a specific format that should be used (always follow journal instructions to the point)