Q:

What are the powers of mayors in the US?

I know very little about US government and state system and only in general. 
I heard that, for example, US town mayor also rules town police and can change taxes. Is it true? The fact about ruling police impressed me. In my country, for example, police is in under federal commandment from minister, and it is logical, I think. 
If it is, isn't this overpowering? Because he becomes a small king if it is so. Or are there some limits?
For example, we have a small town in Texas. What can its mayor do? How much does it differ from state to state if any?

A:

First off, to clarify: the executive of a town/city in USA is typically called a "mayor", not a "governor".
The powers of a mayor may vary from state to state and city to city, BUT:

Yes, mayors usually control local law enforcement (police).

This is not a new concept - local law enforcement was a feature of British system - although at a regional, rather than town, level (remember the infamous Sheriff of Nottingham of Robin Hood fame). City authorities had local law enforcement responsibilities in many cities in late medieval and early modern Europe. 
Overall, the political system - which heavily features separation of powers - is designed in USA on a fractal level (meaning, it looks the same for Federal, state and municipal level) - each level has a separation of powers (executive, legislative, judiciary) and law enforcement is (almost) always a function of executive branch in USA; be it FBI in federal level or state police on state level or town police in the town.

No, mayors do not (typically) control almost any tax levying:

A vast majority of non-property taxation in USA is levied on the federal level (federal income tax and corporate tax, gas tax etc....); some on state level (especially sales taxes and state income taxes); and a very very minuscule amount on local level (some - larger - cities like NYC may have a municipal income tax but most smaller cities do not. Some have sales tax, but most use state sales tax).
The only taxes that are mostly at municipal level are property taxes - and these are only paid by people who own property (land and buildings/houses).
Importantly, as with most other taxes in USA, they aren't levied by the executive branch - but by legislative (in the municipal level, usually City Council); following the same exact fractal version of separation of powers discussed above. What the Mayor is responsible for is to spend the collected taxes; in a way decided on by City Council. In other words, Mayor runs the police, but City Council votes on Police budget; as well as laws governing what Police does.

So, no, a mayor cannot be "like a king", no more so than a President can (far less so, since aside from having their powers limited by separation of powers, they are ALSO limited by what state and federal level imposes).
However, they can amass a ton of effective power, in theory, especially if corrupt. For most striking examples, see:

Tammany Hall in New York City (although one can quibble that it was a general political machine that owned the mayors, not the mayor that owned the political machine)
Corrupt mayors of Chicago (especially Daly I suppose, but I know relatively little of detailed Chicago history)

A:

It depends...
Even within a single state, there may be different governing structures for different cities.  When a city gets large enough, it gets a "charter" that describes the nature of the government.  Different cities' charters can be very different.
Most cities have either a "strong mayor" or a "weak mayor" government.  In a "weak mayor" government, the mayor is pretty much a city-wide city councilor.  The weak mayor form usually delegates most management to a strong "city manager" that is responsible to the "city council" (the city's legislature).
Though Wikipedia says that "Most major and large American cities use the strong-mayor form of the mayor-council system", that's not always true.  Dallas (a very large city in Texas), uses a weak mayor system with a strong city manager.
One other key difference between US cities has to do with how the city councils are elected.  In some cities, each councilor represents a neighborhood.  In other cities, the council is elected "at-large" (every citizen votes for all positions on the council).  There are also hybrid models where some positions are by neighborhood and some are at-large.  Since this is the US, the racial history of the town can figure into how councilors are elected (by-neighborhood councilors are more likely to include minority representatives)

A:

One important thing to understand is that power is decentralized in the American system. Power is pushed to lower levels as much as possible. The federal, national government has very limited powers. States and cities have more power.
Because it is not centralized, there is not one answer to your question. There are a variety of levels smaller than the state, and in each state they work differently. New York has counties that contain towns that contain villages. Virginia has counties and cities - cities are not contained in counties. Connecticut has no counties at all, only towns. California has counties that contain cities, but some areas are in a county but not in a city (unincorporated areas).
The leaders of these levels of government vary as well. I would point out here that another important concept is separation of powers. At the national level there is executive (President), legislative (Congress), and judicial (courts). These smaller levels of government also have separation of powers. In smaller cities there may be a council of 3-10 people who are the executive, rather than a single Mayor. And the legislature can be ~300 elected citizens, or even the entire voting population. So many citizens do not live under a mayor.
And separation of powers prevents any one person becoming a king. Taxes are not changed by the mayor, but by the local legislature. Cities usually only collect one kind of taxes, taxes on land/property. The city only provides a few services. And a tax increase often has to be approved by voters first, limiting how much they can collect. The police do work for the mayor, but their budget is set by the legislature, and their conduct is reviewed by the courts. In almost every city this works well enough to prevent major corruption or autocracy.
In some larger cities the city provides many more services, and collects more tax dollars. With increased money and power, the chances for corruption increase. That is why in several historical examples you see a "small king" developing, like in New York City and Chicago in the 20th century. 
The conduct of mayors is reviewed by higher levels of government, so it is possible for truly corrupt mayors to be arrested by the state government and go to jail. New York and Illinois have both sent high level politicians to jail.

