A Path is a closed vector path or shape, used to cut out an image in
current image editingsoftware like Adobe Photoshop. Once the path is
created, clipping path is then applied on it so that anything outside the
path will be omitted when you place the file into Illustrator, InDesign or
any web developing software and inside the path will remain same as before.

Clipping Path is necessary when we use any photo/photograph/image to
prepare a design, such as brochure design, magazine design, web design,
leaflet and many more. For example, I have a photograph with background
color gray and need to be used in the front page of my catalogue. I must
remove the background of that perticular image since my Catalogue color
is green and have some otehr design. To do so, I will need to create path
 around the image and then convert it into clipping path. And then, when
I use it in Illustrator, InDesign, Dreamweaver or any other program, the
image background will disappeare, mean no background (transparent).
It will show only the image which was inside the path. Look at the images
below to get an live view.

Before After 

1. Open an image in Photoshop you want to use for your clipping path tutorial (go to the file menu in Photoshop and select open, from the open dialogue box choose the file you want to use and then click open. 

2. Once you have got the file open in Photoshop, select the "Pen Tool" from the tool box (if you can't see the tool box, go to window menu and select "Tools"). Now, make sure that you have selected "Paths" from the pen tool menu bar which is located underneath of the main Menu bar. 

3. Now click on the image edge and drag the mouse pointer towards the next step and release the mouse. After an appropriate distance, again click on the image edge and continue in the same manner until you have completed moving around the image and finally see this "o" symble. This "O" symble means you are going to close the path. Click on it (starting point) in order to close the path.

4. Once the path is completed around the image and its closed properly, go to path pallete (window menu/ select "Paths"). Double click on the "work path" and press enter to save the path and the path name will be changed to "Path-1" instead of "work path".

5. Go to "pop up" menu from the Paths palette and select "Clipping Path...". A dialogue box will appeare asking to choose your desired path and Flatness. Select the path you want to Clipped and Put the flatness you desire and click "OK". You are now almost done. Flatness is the desired amount of pixel you want to make blur the image edge. The more amount you use in Flatness the more image edge will be blurish. In general, leave the Flatness box blank (0) and the image edge will be 100% sharp which is most of the time appropriate.

6. You can now save the file in PSD or TIFF formate and ready to place into the program you are working with for your design. This time you will get the image without the background. Thats all about clipping path.

Here are some other tutorials you can try..

Learn how to do Masking in Photoshop
Learn how to Retouch Image in Photoshop
Learn how to Edit/ Enhance Photo in Photoshop
Learn how to Create Shadow in Photoshop
Learn how to Create Vector Image from Raster

Hopping the above tutorials help you.

Regards,
Clipping Path Specialist