If you have a stain that is deep into the fibers or covers a larger area, you're going to need to replace the damaged section. Hopefully you still have some remnants of carpet that you saved when it was originally installed, but if you don't, you can still get a replacement section from the back of that closet.

A couple of points
When you cut out the damaged piece of carpet, save the piece of underlay that comes out. You will need to put it back into the hole to make sure the height of the patch is the same as the original carpet. If the underlay itself was damaged, you can use a piece of material like burlap as a substitute.

The shorter the pile of the carpet, the harder it is to make a patch invisible. Unfortunately, patching a carpet is unlikely to make it look as good as it did originally, but it will still look a lot better than a big stain or burn.

Step 1: Remove the Baseboards, if Necessary

If the patch is to be applied against a wall and the room has baseboard molding, your two options are: a) remove the baseboard and reinstall it after laying down the carpet patch, or b) leave the baseboard in place and install the tack strip 1/4" away from the baseboard as described in the next step.
Step 2: Attach the Tack Strip

If the area you're patching is up against a wall, you'll need to put down a tack strip which has "teeth" that hold the carpet in place. Nail the tack strip along the wall 1/4" away from the edge of the wall or baseboard.
Step 3: Cut the Carpet Pad

Cut a piece of the foam carpet pad to fit the area to be patched. The pad should be cut to fit inside the tack strip, not on top of it.
Step 4: Determine the Nap Direction

Next, check the carpet nap to make sure it follows the direction of the existing carpet. Run your hand back and forth over both pieces to check.
Step 5: Cut Out the Patch

Once you have determined the nap direction for the scrap, use a utility knife to cut a piece to size and fit it into the area.
Step 6: Apply the Seam Tape

If one of the edges of your patch is against a wall, attach the other three edges first using the process described in this step. The last edge will be attached to the tack strip in the next step of the project.

Lay down the seam tape just under the seam and use the carpet iron (Image 1) to activate the seam tape and bond the edges together. Then, use a carpet tractor (Image 2) to press the carpet down onto the seam tape to reinforce the bond. Afterward, position something heavy over the area to help set the seam.
Step 7: Stretch the Carpet Over the Tack Strip

When all the seams are secure, use a knee kicker to stretch the carpet onto the tack strip. Always work down the length the wall to the innermost corner. Use a hammer to push the carpet down onto the teeth of the tack strip.
Step 8: Trim Away the Excess

Trim the excess with the carpet trimming tool, then press the edge of the carpet underneath or against the drywall. Reinstall any baseboard molding that was removed.

Try see more of this tips here: