Cleaning Up Dog Poop

This method is for non-porous surfaces, like linoleum.

Clean up as soon as possible.
Put your puppy in a place where she CAN'T see you cleaning up.
Remove urine with paper towel and pick up stool with toilet paper so you can flush it down the toilet.
Wet the area with a homemade or commercial cleanser.
Always clean from the outside toward the middle to avoid spreading the mess.
Rinse twice, changing water each time.
Spray the spot with an odor neutralizer.
Block access to the spot until the product is completely dry.

Method for porous surfaces, like carpet.

Clean up as soon as possible.
Put your puppy in a place where she CAN'T see you cleaning up.
Sprinkle bicarbonate or absorbent flakes, wait for the liquid to be absorbed then do the vacuuming, or apply a paper towel to absorb the urine. Pick up the stool with toilet paper so you can flush it down the toilet.
Always test the cleaning product first on a hidden area of the carpet.
Wet, but don't soak, the soiled area with the homemade or commercial cleaning product and wait a couple of minutes. Gently blot the surface.
Always clean from the outside toward the middle to avoid spreading the mess.
Spray the spot heavily with an odor neutralizer to make sure it reaches the underlay and even the floor.
Block access to the spot until the product is completely dry.

Details

Step One
The longer it takes you to clean up, in particular if your puppy has soiled on a carpet, the deeper the urine will sink in and the harder it will be to get rid of the smell.

Step Two
There are two reasons why your puppy shouldn't see you cleaning up her mess.
First, you don't want her to see you as her maid. Furthermore, she may decide the next time to clean up her mess by eating it. By the way, the most effective way of stopping this degusting behavior is to always pick up your dog's feces right after she has eliminated.
The second reason is that you probably won't be in a very good mood. If your puppy sees you in that frame of mind, she may decide to hide the next time she needs to go because she'll be afraid to eliminate in front of you. Needless to say that this can only slow down her housetraining progress.

Step Three
You can also pick up your puppy's stool with a plastic bag and dispose of it.

More Cleaning Tips
Your puppy may have eliminated somewhere without you ever being aware of it. One very useful tool that you can rent or buy  is a portable black light. The UV light will help you locate stains from urine and the hidden sources of odor allowing you to clean them up thoroughly. Don’t hesitate to clean a larger surface than the visible soiled area. Your puppy may very well have walked in his urine and "bring it" on a couple of feet.
By the way, don't use a sponge to clean. It is very difficult to completely wring all the liquid from a sponge. Consequently, there's a big chance that you'll spread the urine that you are trying to clean everywhere!
If you've washed a spot more times than you can count and your puppy continues to eliminate there, it's now more of a habit for your puppy than anything else.
The first thing you can do is to use a repellent. Just spray the spot and see how your puppy reacts to it. It may very well solve your problem.
The second solution is to supervise your puppy very closely. Maybe you thought she was completely housetrained and allowed her too much
freedom. Go back a couple of steps. Keeping her in her domain one more week may also help break the habit and reinforce the use of her elimination zone or toilet stop.

Conclusion
Remember that cleaning up is a vital part of your puppy's housetraining progress. If you do it conscientiously, you dramatically reduce the chance for your puppy to have an accident. Although this situation is temporary, always have cleaning products on hand. You never know when your little genius may get into trouble!