The point where the two asymptotes cross is known as the center of the hyperbola.
Standard forms
There are two standard forms of the hyperbola, one for each type illustrated above. Here is a table giving each form as well as the information we can get from each one.
Note that the difference among the two forms is which term contains the minus sign. If the y term contains the minus sign then the hyperbola will open left & right. If the x term contains the minus sign then the hyperbola will open up & down.
We obtained the equations of the asymptotes using the point-slope form of the line & the fact that we know that the asymptotes will go from the center of the hyperbola.