Can a standard outlet be protected by a GFCI outlet?
Absolutely possible, if done correctly.
First, the simple test

Make sure (nightlights or any other small devices) that both receptacles work (i.e., GFCI and regular in kitchen; or GFCI and regular in bathroom)
Press the TEST button.
Verify that BOTH receptacles have lost power.

If only the GFCI receptacle loses power then either the regular receptacle is not wired to the GFCI correctly, or it is on a different circuit altogether. In both of these locations, it is important that all receptacles be GFCI protected, so this is an important thing to test (and to fix if needed).
How it works
A typical GFCI receptacle includes two sets of screw terminals. One is labeled LINE and the other is labeled LOAD. The LINE side is connected to the incoming hot &amp; neutral wires. The LOAD side is connected to any downstream protected receptacles. The GFCI electronics detect any imbalance on the attached receptacle and on anything connected using the LOAD terminals.
What if it doesn't work?
If the regular receptacle doesn't turn off when the GFCI is tested then there is a small possibility that the GFCI is defective. But far more likely is that the regular receptacle is connected on the LINE side instead of the LOAD side. That may be double wires on screws (allowed in some cases, depending on design of the receptacle) or by connecting incoming hot, GFCI LINE hot and a third wire to the regular receptacle together, and the same connection with the neutral wires as well. If that's the case, you just need to move the wires going to the regular receptacles over to the LOAD side of the GFCI receptacle.