Both заявка and заявление appear to mean application. In which, if any, situations is one correct and not the other?
What's the best way to express job application?
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I am not entirely sure what English "job application" means. If it means you send a CV seeking a job or a position, there is no equivalent Russian term, you have just describe this as sending CV for a position.
Заявление о приёме на работу is the official part. This happens when you are already approved for the position. It is just a part of the official paperwork, describing your official intention.
In general, заявление is a very official term. It always means something properly signed and addressed.
Заявка is more like a request or a booking.
"Заявление" doesn't always mean "application". It can also mean "a statement". For example: "я катергорически не согласен с вашим заявлением!"
In cases when it does mean "application", "заявка" is kinda similar, but not quite the same. "Заявление" is more closely associated with the actual official paperwork that the application involves. It is also more of a request, while "заявка" is kinda closer to a statement of intent.
For example, one would, usually say "я подал заявку на участие в соревнованиях", not "я подал заявление ...", because you are not asking someone to let you compete, rather just notifying them that you intend to do so. On the other hand, if you would like to get a visa to visit Russia, you would send a "заявление" to the embassy, not "заявку".
The examples in one of the other answers (like "заявка на кредит" or "заявка в санаторий") are actually incorrect (those would be "заявления"). "Заявка в газете" is wrong too (it is "объявление").
One way to look at it is that "заявление" is asking somebody to do something, while "заявка" is notifying someone about something you intend to do (even if you actually need their permission to do it - that's where the boundary with "заявление" becomes kinda fuzzy).
Also, another possible meaning of "заявка" is "claim": "это серьезная заявка на победу" == "this is a serious claim to victory".
Заявление is formal/official request from natural person to a corporate entity, department, agency, etc to do or accept something, e.g. "заявление о выдаче паспорта" is a request for new passport, or "заявление на отпуск" is request for vacation. Bureaucracy stuff.
"Заявление о приеме на работу" - job application
Заявка is an announcement of intention, very similar to request. E.g.:
"заявка об оплате" is a bill from a company for their services (announcement that you have to pay / request for payment)
"заявка в санаторий" in communism UdSSR employers offered their employees to visit a sanatorium (kind of spa hotel).
"заявка в газете" is announcement or advertisement in a newspaper (like "sell my mixer, alsmost new, call me maybe")
"заявка на кредит" annouce, that you want a bank credit / request for bank credit
"заявка на участие в конкурсе" announcement, that you want to join some contest / competition
Actually every request is kind of announcement, that you want something. But "заявка" is more "i want to ..." and not "could i do ... ?"
As I understand, term job application means first step in job seeking process, when a person, who needs the work, sends a resume. There is no term for this in Russian. On russian job seeking sites, like hh.ru or others, this action is called "откликнуться на вакансию".
In colloquial speech, you can use "отправить резюме". For example:
Вчера наконец-то нашел отличную вакансию! Отправил резюме сразу же, теперь жду ответа.