How can we fix this extra exposure at the bottom of the vase in the attached picture
The bright areas of that image (as it appears in my browser) are clipped, that is they are pure white (#FFFFFF). Here's your image with a histogram in the lower right corner:
See how the histogram "hits" the right side there? That shows that it is clipped, or "whited out". That means there is no information there in those white areas with which to work, no image data to recover.
If you are taking the picture you should learn to "expose to the right", you can adjust the exposure compensation to make the picture darker so that the histogram doesn't touch the right side. Then it's (usually) easy to lighten the dark areas of the picture.
Use the burn tool:
Just paint the areas where you pretend to lower the exposure. Be sure to adjust the exposure to highlights in the tools option, since the default is midtones. You can also play with the exposure to soften the effect.
In alternative, you can also create a curves/levels/brightness adjustment layer and mask the affected area:
Just select the adjustment you want from the adjustment panel (or near the new layer icon if you're using something below cs4) and adjust the exposure for the light area of the image.
Then click on the layer mask (the white box in the layer list) and select a brush (with the black color selected) and paint the areas you want to remove.
Edit: With the above picture however it seems the image file doesn't have the necessary information to recover the details lost by overexposure. The only way is to retake the photograph. If you have a raw version of the file, work from there instead as these files capture much more information of the light captured by the sensor.