From my understanding you have both sockets below in your apartment:
It sounds to me that your apartment must be quite an old building, and that it hasn't been 'grounded'. The reason for the three pin socket could be that a two pin one could have been in it's place and became faulty, and someone one day replaced it with a three pin type and didn't bother grounding it.
Your surge protector actually seems to have a ground tester inbuilt. It is possible to test for grounding with a multimeter, however it's much safer to use a dedicated tester like the one pictured below (which you can buy from most major hardware stores).
In Australia, we call ground 'Earth'. Earthed outlets have an extra pin that's wired to a stake in the ground (the dirt somewhere around the perimeter of your building, usually near the fuse box). However in line with this, there's a device called a residual-current device (RCD http://en.wikipedia....-current_device), which in the case a small electrical current is sensed flowing down the line (ie. in the case of a fault) will trip and disconnect all power. RCDs are a requirement by law in Australia (however there's still quite a few people living in older homes (pre-1930s) who haven't bothered upgrading), and on many instances have been life-savers.
There are two types of electrical devices:
- Earthed: those with a metal case/body connected to earth, so in the case there's a electrical short inside, current will flow to Earth and trip the RCD, protecting the user, and
- Double-insulated: usually devices with a plastic casing, that in the event of something going wrong will be of no harm to the user who may be holding them (ie. most plug packs and some small power tools).
If you test that your three-pin outlets are not grounded, I would contact the apartment owner to organize proper grounding. As mentioned above, it could be a life-saver one day. However if you only use double insulated devices (ie. TVs/laptops/phone charges, items that have a Class II symbol on them (see http://en.wikipedia....pliance_classes), and never pull things apart), then grounding's not really required (however I'd highly recommend it anyway).
Your surge protector will still fully protect against a surge regardless of grounding (it contains electronic components internally for this).
Regardless of whether grounding goes ahead or not, it is important to realize that the electricity coming from a wall socket can be lethal. SAFETY FIRST.