The authoritative answer is at Way to Russia, but I'll summarize for posterity.
- Whoever you're staying with, be it hotel or apartment, is supposed to take care of registering you.
- They can take care of the registration at any police station or post office, you do not need to be present.
- When it's done, they'll get a piece of paper for you that confirms registration, and all is good.
If you do not register, and you're stopped by police, and it's been over a week since you arrived in Russia, you'll be fined (€50-100). In theory, you can even be deported and blacklisted for returning for 5 years, but this is very unlikely.
In your case, I'd get in touch with whoever is renting you the apartment and ensure they sort it out, the magic word is propiska (registration). It's in their best interest to do so as well, since they can also get fined even more heavily for having unregistered foreigners staying with them.