Incenter is the intersection point of bisector lines of the angles of the triangle
Circumcenter is the intersection point of the sides perpendicular bisectors
Ortocenter is the intersection point of the height
Centroid is the intersection point of the medians
The average of the coordinates gives the centroid.
The other points are less trivial to find
Incenter can be found using the formula here
http://www.mathopenref.com/coordincenter.html
Circumcenter
P has coordinates
<br />
x_P=\frac{{x_A}^2 {x_B}-{x_A}^2 {x_C}-{x_A} {x_B}^2+{x_A} {x_C}^2-{x_A} {y_B}^2+{x_A} {y_C}^2+{x_B}^2 {x_C}-{x_B} {x_C}^2+{x_B} {y_A}^2-{x_B} {y_C}^2-{x_C} {y_A}^2+{x_C} {y_B}^2}{2 (-{x_A} {y_B}+{x_A} {y_C}+{x_B} {y_A}-{x_B} {y_C}-{x_C} {y_A}+{x_C} {y_B})}<br />
<br />
<br />
y_P=<br />
\frac{{x_A}^2 ({x_C}-{x_B})+{x_A} \left({x_B}^2-{x_C}^2+{y_B}^2-{y_C}^2\right)-{x_B}^2 {x_C}+{x_B} \left({x_C}^2-{y_A}^2+{y_C}^2\right)+{x_C} ({y_A}-{y_B}) ({y_A}+{y_B})}{2 ({x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B}))}<br />
Orthocenter
O has coordinates
<br />
x_O=\frac{{x_A} ({x_B} ({y_B}-{y_A})+{x_C} ({y_A}-{y_C}))-({y_B}-{y_C}) ({x_B} {x_C}+({y_A}-{y_B}) ({y_A}-{y_C}))}{{x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B})}<br />
<br />
y_O=\frac{{y_C} (-{x_A} {y_A}+{x_B} {y_B}+{x_C} ({y_A}-{y_B}))+({x_A}-{x_B}) (({x_A}-{x_C}) ({x_B}-{x_C})+{y_A} {y_B})}{{x_A} ({y_B}-{y_C})+{x_B} ({y_C}-{y_A})+{x_C} ({y_A}-{y_B})}<br />
It can be quite hard to work it out...
Graphically it can be seen as in the following picture