'Left-wing' and 'right-wing' refer to styles of politics.
Democracies, autocracies etc are types of government.
The difference is important. Although politics and government are closely linked, they are not synonyms.
Wikipedia gives this broad explanation of the difference between left and right wing politics:
Generally, the left-wing is characterized by an emphasis on "ideas such as freedom, equality, fraternity, rights, progress, reform and internationalism" while the right-wing is characterized by an emphasis on "notions such as authority, hierarchy, order, duty, tradition, reaction and nationalism"
Most people recognise that politics of individuals or parties do not strictly fall to the left or right, but exist on a spectrum. For example, in the UK, the two main parties are broadly on the left and right respectively, but share some common ground. Extreme authoritarian parties such as the Nazis tend to be called "far right" because of how extreme they are in their authoritarian policies and how far away they are from any left-wing policies. As far as I know, the term "far left" isn't widely used. In the UK, the pejorative term "loony left" has been used to mock politics which are so focused on pleasing individuals that they are considered unworkable for the majority.
In countries that are a democracy, this means that the people can vote for a political party to rule, and those parties may be 'left' leaning or 'right' leaning, politically. But even where there is no democracy, both styles of politics can exist. For example, communism is one 'extreme' form of left-wing politics, yet some communist governments like Russia and North Korea arguably also display qualities of authoritarian, extreme right politics.