I can confirm you that it's NOT normal at all. When the charger is on, provided it's the right one and the Tension/Power is correct, the mac should not do that.
If your battery is below 98% (apron.), plugging it to the wall should charge it and then it should switch to "Charged 100% (or 99/98%) and start using the power from the outlet (and not the battery).
THe Not Charging can occur when:
- The battery is faulty (and therefore it doesn't charge) so the board is reporting this to you: "hey, this battery is not charging and it should".
- The Power Adapter is not delivering the correct power (or enough power if the tension is low). This can cause the charging board to cut the battery charging process because there's not enough power to guarantee a proper charge (and failure to do this may result in your battery life being decreased).
- The unlikely event where you've connected way too many devices to your USB/FW/Etc. ports in your computer, and the power adapter can't keep with the power demand. (I've only seen this happen once with a POWERBOOK a long time ago).
Skype, with all the bad things it has, cannot drain your battery like that (unless, of course, it's a high CPU usage like bneely said in the previous answer).
If you restart and keep noticing this, I'd recommend two paths:
1) Create a new blank user and try from there. See if you get the same problem. If yes, then see #2, if not, then it's some software problem and we can move from there.
2) If the issue still persists with a new user after a hard reboot and you've performed the basic stuff, I'd take the machine to an App Store for further inspection. If your power adapter is bad, it can slowly damage your battery capacity (it's the same as if you use a higher power adapter than the one you're supposed to use, yes, it can charge faster, but you're draining your battery).