I know how to use articles "a" and "the". I know how to use "is" and "are". But I don't know when I should omit them. Back in my childhood I was visiting English language lessons (my native is Russian, as well as my teacher's from my childhood). And she was very confused about the phrase "Game over" which I've asked her about because I was obsessed with video games. She told me that the phrase is wrong, and it should be "The game is over". Also a long time ago I've played some spacesim video game and a very nice girl's voice has been vocalizing some of the game events. And I remember such phrases as: "Target detected", "Target destroyed. No new targets detected". No a/the, no is/are. Also if you played Starcraft for the Zergs then you should remember the phrase which sounds upon completion of an evolution of a "building": "Evolution complete". And it's not "The evolution is complete". So, what's the trick? When should I omit a/the, and when should I omit is/are?
Your teacher is right. "Game over" is ungrammatical and intentionally so. Game designers routinely drop articles and linking verbs to keep messages as short as possible. This saves screen space and makes messages easily skimmable.
Games used to have to deal with very limited resources like screen space, and it made sense to try and keep messages short and clear, also to avoid burdening the player with having to read a lot of text. (Imagine "Dear player, we are sorry to inform you that notwithstanding your great efforts, your gaming experience has now been terminated".)
Games have copied the use of common signs in that: they are quickly recognizable, even though they may not contain a full grammatical sentence. "Exit" successfully conveys the meaning of "this way you will find a way to leave the premises", "Ladies" in the same manner makes clear the presence of sanitary facilities for people of feminine persuasion.
The same goes for military commands (from Why is “Consequences inflicted.” not a sentence?):
Native speakers sometimes do say things similar to that. Most of the examples that come to mind are from the military: “Countdown initiated,” “Missile launched,” “Target acquired,” etc. The copula—the "is", "was" or "has been"—is implied but unstated. It’s a minimalistic way of speaking associated with situations where every second counts. Robots in science-fiction stories tend to speak this way, too.
In science fiction this might be a stylistic choice to emphasize the robotic nature of a speaker by imitating the terse style of early computer programs (Item modified, File Not Found).
It might be argued that all of these examples are special cases of headlinese: the abbreviated writing style used in headlines, slogans and street signs.
So, going back to your question.
When should I omit a/the, and when should I omit is/are?
Whenever you need to save space. Be extra careful, as sometimes omission of linking words leads to confusing and/or ambigous sentenses. Keep in mind that the text like this is ungrammatical and might come off as robotic.
- In which situations is it OK to omit articles in short sentences?
- Style of technical warnings. Why is it acceptable to omit verbs?
- Why do newspaper headlines use strange syntax rules?
Side note: there exist a lot of special cases in which articles and auxilliary verbs can be dropped legitimately. You'll need to write a small book to enumerate all of them. I decided to concentrate my answer on the special type of language you're asking about.