The term you are looking for is recall .
The wikipedia article suggest Venezuela is the only country with a national process to unseat a leader by petition.
But the US, Canada, Switzerland and Ukraine have options for removing lessor leaders.
Most notably the governor of the state of California was changed by recall in 2003.
This isn't a country exactly, but it is a very large and important district.
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No, since impeachment means charging a government official with a crime or misdemeanor.
As Wikipedia phrases it : Impeachment is the process by which a legislative body formally levels charges against a high official of Government.
Impeachment does not necessarily mean removal from office; it is only a formal statement of charges, akin to an indictment in criminal law, and is thus only the first step towards removal.
Once an individual is impeached, he or she must then face the possibility of conviction via legislative vote, which then entails the removal of the individual from office.
Doing this by public opinion polls would be ridiculous.
If you mean "allow leaders to be removed from office by low approval ratings", then also no, I can't find any examples of countries that have codified that.
For better or worse, your vote is for a period of about 4 years (depending on your location etc.)
Changing leaders/direction every year based on a whim or a temporary setback/scandal/downturn is probably not a great idea.
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Practically any country with a prime minister allows leaders to be removed by unpopularity.
They aren't "impeached" but just replaced.
The prime ministers could be unpopular with their own party and replaced as party leader.
That happened to Margaret Thatcher for example.
Prime ministers tend to be chosen by coalitions.
A member of the coalition can leave and join the opposition.
Then a vote of confidence can be called, fail, and a new government formed.
The exact procedure of course differs from country to country.
The new government does not necessarily need to include the leading party of the old government.
If a government can't be formed, some countries may have provisions to force a new election.
In the United States, two presidents have been impeached and put on trial.
Neither was removed from office.
In both cases, the impeachment reason was something of a pretext.
Andrew Johnson was deliberately baited into violating a possibly unconstitutional law.
Bill Clinton gave misleading answers to improper questions about his personal life.
So effectively an unpopular president could be removed by impeachment.
They'd just need to find an excuse to do so.
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Yes, by definition all democracies have the function you describe.
An election is a mechanism for measuring a candidate's public approval relative to other candidate's.
An elected official's term eventually ends and they can be removed from power or brought back solely due to their relative approval rating.
Many, but not all, democracies have a function is used to remove an officer from power based on their approval ratings without waiting for their term to end.
These may be called recall elections, although they may have other names in other localities.
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NSBD did a great job pivoting with your question-- Impeachment is a process that can only be invoked following a crime (though, as history has shown, the individual does not always wait for the results of an impeachment proceeding, and may resign preemptively -- e.g. Richard Nixon).
In the case of Richard Nixon, Senate confirmation of the impeachment was all but certain, but this not always the case -- Bill Clinton was impeached by the House (i.e. articles of impeachment based during House impeachment proceeding), but was acquitted during the Senate proceeding.
Numerous U.S. states have procedures for recalling executives.
In fact, the recall (along with the ballot initiative and referendum) are key achievements of so-called progressive reform in the U.S.
They are also examples of direct democracy.
While California's recall of Gray Davis is one of the more well-known examples of the use of a recall, eighteen U.S. states have recall provisions.
To add my "two cents" here, it is a good thing that presidents can not be removed due to low opinion polls for three main reason: 1) polls are an imperfect measure of popular opinion
so it's possible a poll does not precisely reflect "true" popular opinion); 2) unpopularity is a reflection of a president's personal and political shortcomings, and these shortcomings should be addressed through political means (elections, but also other features like federalism and checks/balances between the executive and other branches); and 3)
[this one is a bit more of a stretch] like
the Judicial Branch sometimes makes decisions contrary to popular opinion, so does the President sometimes have to make unpopular decisions.
For example, it would be inopportune for a President to be impeached during a war or natural disaster as opposed to afterwards as a reflection of their incompetence, ineffectiveness, etc.
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It seems logical that a leader be removed if a high majority of people no longer want the leader in power. Are there any countries that allow leaders to be removed from power if their approval ratings are extremely low?
Do any countries allow leaders to be impeached by low approval ratings?
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<impeachment>
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By definition, all democracies have this function.  Practically any country with a prime minister allows leaders to be removed due to unpopularity.   Impeachment means the official needs to be charged by a crime or misdemeanor but a leader can be removed by unpopularity.
Impeachment means the official needs to be charged by a crime or misdemeanor but a leader can be removed by unpopularity.