Hello, I wanted to know that how researchers in mathematics keep updates in their field of interest.
(original question by Rahul Gupta)
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There are several ways, and it depends on the precise meaning of 'in their field of interest'.
With a narrow definition of this an important and classical way to keep up to date is personal correspondence, people in the same field often know each other and thus people actively inform people of which they know that they are interested in progress on a certain problem directly. More recently, some people also use blogs for announcments of latest progress.
With a less narrow defintion: As mentioned in a comment, there is a large preprint server arXiv.org where many researchers post their preprints. One can subscribe to keep daily emails on new postings in specific subcategories.
Attending (specialized) conferences is also an relevant way to keep informed.
Finally, most fields have specialized journals, and one can browse them to see whether one sees an article one wishes to study in more detail.
So, personal communication, preprints, (specialized) conferences, (specialized) journals; and more recently blogs and personal websites, is what comes to mind at the moment.
The most effective way for me is to give talks, go to conferences and workshops, and talk to people. E.g., I gave a talk at Stanford on Friday, and learned much I didn't know about some "aspects" of recent research in number theory from Brian Conrad, while walking back from lunch.
Regarding the internet, there are specialized webpages that sometimes track news for various research communities, as discussed recently here. I started reading mathoverflow a few months ago, and currently for me it is bar far the best online way to find out about current events in math research (at least in my area -- number theory). It's just stunning the number of new results and links to key papers I've found on mathoverflow.
Recently one of my students asked me "Why didn't I know that the Sato-Tate conjecture had been proved?" I responded "Do you read mathoverflow?" and he said "no", and it's what I suggested he start doing (instead of reading slashdot and new york time, say). I find out about so much interesting research here. The arxiv is not as useful, since there's no discussions, rankings, etc. Mathoverflow successfully leverages "crowd sourcing" in a way that works for research math, at least better than anything else online I know of.
One thing I would add to unknown's answer is RSS feeds. All the electronic resources mentioned: blogs, arxiv, journals, mathoverflow---they all have RSS feeds. Instead of compulsively looking at blogs each day for rare updates, or browsing arxiv all the time, I just go to my feed reader (mine happpens to be Google Reader).
Also some pages of interest may not have feeds. For those you can use a change detection service like http://www.changedetection.com/ which can send emails to you when a webpage changes. I for example use this to monitor the Assoc. of Symbolic Logic's list of conferences. I also use it to monitor certain publication lists of colleagues on their websites.
This can be a lot of information overload, but if one develops the habit of quickly scanning through the items, ignoring anything not of interest, then it isn't bad.
The same as everyone else: arxiv, conferences, in my case "Number theory web", talking to friends and colleagues, etc. But I've come to realize that all of this keeps me updated about a very small portion of mathematics. To remedy this, once every couple of months I walk down to the math periodicals section of our library and spend an hour or so browsing the most recent issues of good journals: read abstracts, scan keywords, etc. I somehow find this very helpful; granted, by the time papers appear in print, they are for the most part outdated, but this still gives me a sense, though with some delay, as to what's going on in the mathematical world. In the same vein, I find the Notices of the American Mathematical Society very helpful.
I use Google Scholar Alerts. Alas, they are far from optimal. First, Google has an epically flawed system of recognizing the names of the authors from a paper (so a paper about cyclic homology will more often be shown as authored by "C. HOMOLOGY" rather than by its actual author). Second, they do a bad job at filtering. My "Witt vectors" scholar alert spews out loads of epidemiology and medicine texts due to Witt being a not-too-rare surname and "vector" also meaning a way a disease is spread. (This even got some of the Google Scholar Alerts I receive automatically sorted into the Spam folder by Google's very own GMail spam filter. Apparently the culprit was a paper containing "penis" in its summary.)
Still, out of all ways to obtain digests of new research activity, this seems to me the best one at the moment.