how to fix gear ring that over-jumps I am pretty new here, and I am not acknowledged with the bicycle terminology in English, so this topic might be a duplicate. Sorry in advance.

So, when I put more pressure while pedaling, the chain over-jumps (some teeth of the gear ring are being missed). Upon observation I think the problem is in the front gear rings (they're 3), but I am not quite sure. When the chain is on higher gear (smaller gear ring) on the back ones - it jumps more that when I am on a lighter gear (bigger gear ring). 

Any suggestions what can be the problem and how can I fix this ? 

P.S. the teeth on the gear rings seem to be fine and not waster or smth.
Your issue sounds like a worn chain/cassette.  Take the bike to your LBS (Local Bike Shop) and have them look at it. You can also check yourself to see if the rear cassette teeth look like sharks teeth, that's generally a good indicator of wear.  Here is another post that may help you.  

Generally skipping teeth is a sign of a worn cassette/chain, but it could also be adjustment issues from the derailleurs.