How do you organize/read research papers

Read the Abstract 
read the Results & conclusions 
if ( interesting and relevant)  {
        copy into Mendeley (http://www.mendeley.com/)
        read the rest of the paper.
}

Even I am facing the same problem , but what I usually do is , read first the asbstract ,their main contributions and some of the mathematical equations and the conclusions. If this answers some of my questions, I usually read the whole paper.This may save your little bit of time.

Here is a link for how to read and organize scientific paper.

I hate to add an answer to an otherwise well answered question, but one thing to remember about reading research is that there is a large number of crap or uninteresting papers being published every day.

The reason for this is simple -- publish or perish -- but as researchers, everyone suffers because the signal can get lost in the noise.

This means you must be vigilant when reading research. Can't work out what's going on after reading the abstract? Don't bother. The results are at best incremental using a weird evaluation metric? Don't bother. Very poor communication skills? Don't bother. Paper from 2000 in an active field, but has no citations? Read a little, but don't stress too much. It may sound a little pretentious, but the reality is that we have too much to do to waste time reading a paper from a conference that the authors just put in to bump up their quota.

It's also important to keep track of all the papers you ignore, and why you ignored them. While Dov only puts relevant papers in Mendeley, I put everything in there and try to keep notes on most of the papers (I've been a little slack, but it's a low priority task at the moment). I do this because I kept finding myself getting halfway through a paper and realizing I've read it before.

Here is my personal workflow to complement Dov's:

read the Abstract 
read the Results & conclusions
find the plots with the test set curves / benchmarks
try to dig in the text for info about the number of samples, features, runtime and memory usage
if (interesting and relevant and scalable)  {
   save a copy in dropbox
   if (2 columns layout pdf) {
       paper2ebook the pdf
   }
   send pdf to my kindle for reading when bored in public transportations
   mentally add to my list of stuffs to implement in a WIP pull-request for sklearn
   (or more frequently try to convince someone else to do so)
   (and often, completely forget about it because of paper overload...)
}

paper2ebook is a tool to make 2 columns papers somewhat readable on mobile devices.