Electrical – Why is replaced outlet dead

There are only 2 correct and safe ways to add a 3-prong outlet where formerly there was only a 2-prong one*:

• EDIT: now allowed in Code since this answer was written: Retrofit just a ground wire, using proper wiring methods. There is no need for the ground wire to follow the same route as the power cable. Grounds can be shared: You don't need to homerun back to the panel if you can reach another grounded circuit with the same size or larger ground wire also coming out of that panel.
• Safe way #1: run a new cable with an appropriately sized ground all the way back to the breaker panel.
• Safe way #2: Use a GFCI outlet (with the "test" and "reset" buttons), which will cut off power to the outlet if it senses an imbalance of current. (EDIT: As @EricLippert correctly points out in the comments, you can also replace the circuit breaker with a GFCI breaker instead of using a GFCI outlet.)

There are many wrong ways to add a 3-prong outlet, most of which are dangerous and some of which are hard to spot or test for:

• Wrong way #1: Using a 2-prong to 3-prong adapter without properly grounding the tab (common but obvious).
• Wrong way #2: Replace the 2-prong outlet with a 3-prong one, but only hook up the black and white wires (this is hard to see unless you open up the outlet box, but an outlet tester with reveal the problem).
• Wrong way #3: Replace the 2-prong outlet with a 3-prong one and short together the neutral and ground wires (this is probably the sneakiest because it will test as OK with an outlet tester but is just as unsafe as #1 and #2).
• Wrong way #4: Replace the outlet with a 3-prong one and connect the ground to a random piece of metal (the outlet box, a pipe, metal cladding on an armored cable). This too may check out as OK with an outlet tester but is not safe.

*There are actually a few other ways to ground an outlet safely, e.g. with armored cable with a bonding strip or a grounded box plus an outlet that allows grounding through the strap. But those are both pretty uncommon.

Note that if you can't put in a grounded outlet, a gfci outlet is an acceptable alternative for safety. I've done that at several places in my old house. When installing without a ground, there's a sticker that should be applied essentially saying "yes, we know there's no ground, don't panic if the tester tells you so but also don't expect to use anything plugged in here as a ground."