I want to post some short stories on my blog. But I am not sure about the copyrights of those stories. Because those stories were available in many places online. Can I publish them on my site or should I ask for any permission?
You cannot post any copyrighted material for which you do not have copyright.
But, you can provide links to those materials from your blog.
If you really need to publish, then better ask for a written permission from the original author or the person who holds the copyright.
Just because something is available in multiple places on internet does not imply that your site too can do the same without permission.
If you're looking for stories that you can safely publish online, there are a few good resources for you. @Vivek is right that you definitely cannot publish works to which someone else holds the copyright. It's always safer to try and find a reliable source where those works are already published and simply link to them. That way, if it turns out there are copyright problems, your site won't get in trouble.
However, there are many stories that are in the public domain because their copyright has expired. Project Gutenberg is my favorite resource for public domain books. You can freely read and download any ebook listed on their site, and as most of their books are no longer bound by copyright, it is legal to reuse those published works as you desire. I would, however, recommend reading the entire Project Gutenburg Licence before reusing any of their works.
From the Redistributing section of their licence (emphasis added):
If an individual work is in the public domain in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed.
Note that this is restricted to the United States, as international copyright laws may vary. They also have a few caveats for use that they would like you to follow:
Of course, we hope that you will support the Project Gutenberg-tm mission of promoting free access to electronic works by freely sharing Project Gutenberg-tm works in compliance with the terms of this agreement for keeping the Project Gutenberg-tm name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg-tm License when you share it without charge with others.
Note as well that not all of the works hosted on their website are in the public domain.
Those works will be marked at the beginning of their files on their webpages, so you should keep an eye out for that.
The most important thing to remember is that you need to avoid infringing on any copyrights. For each story you want to publish, you need to find the original author and ascertain whether his works have been released into the public domain within your country or not.
Most countries say that works enter the public domain 70 years after the author's death. If the story is more than two hundred years old, it's safe to say it's within the public domain. Stories from the Bible, for example, are going to be in the public domain, but exactly reproducing a specific newer translation might not be.
You might also want to check out this question and its answers, which explains a lot about works that are in the public domain.