From this...

To this...

Steps
Understanding the Rounded Rectangle tool
You can pick the Rounded Rectangle Tool from the left tool menu.

One important setting for this tool is the radius. That size will determine the radious of your rounded corners. You can choose the size you want.

The more bigger the radious is, more rounded will be your corners (see image below for some examples)

Using the Rounded Rectangle Tool to make our rounded corners
Now, time to use it. The color isn't important here, we just want to make a rounded rectangle shape and then select it.
1.-
First of all, copy your image, open a new document and paste it (never work with the original image file to avoid replacing it unintentionally).
So, with the Rounded Rectangle Tool selected, click on the top left corner of your document and drag it until the bottom right corner.
Remember that you can also change its size using Cntrl + T and you can move it using the Move Tool (V or
)
You'll have something like this:

2.-
Now, we're going to select the rounded rectangle. To do that, Ctrl + click on the layer's thumbnail.

You rectangle will be selected now.

3.-
As we only want the selection, not the white rectangle, we're going to delete it. Click on 'Shape 1' layer and drag it to the little bin icon.

You can see your complete image, with a rounded rectangle selection over it.

4.-
Now, we could inverse the selection and delete the part we don't want, but we are not going to do this because that would modify the original image.
So, instead, we'll simply add a Layer Mask. Click on Add Layer Mask on the layers tab.

This way, using layer masks, if you don't like the rounded corners or you want to go back to the original image, you will only have to delete the layer mask.
Our final result
And done! We have our image now with pretty and nice rounded corners.

Changing background color
Let's say you want to use your rounded corners image in a website with a dark background. To change yours, just select your background layer (see image below)

Choose a color and pick the Paint Bucket Tool (G or
) and click on your document. 
That's it?
Yes, that's it! It's as easy as it seems to be.
This is my newest tutorial, I hope you all like it.
If you have any question regarding this tutorial, don't be afraid to ask and I'll do my best to answer it.