Depending on the context, the 'centre of a triangle' can be thought of in a number of different ways:
First, by constructing perpendicular bisectors of all three sides.
Second, by constructing angle bisectors at all three corners.
Third, by joining up the midpoint of each side with the opposite corner.
Fourth, by dropping a perpendicular line down from each corner to the opposite side (or to where it would extend to if outside the triangle itself.
In each case, the three lines produced will cross in the same place, giving four alternative centres to the triangle. Try moving the corners of the triangle around and see how it affects the centres.

The perpendicular bisector method (shown in red) gives the centre of the circle which passes through every corner (known as the circumcircle).
The angle bisector method (shown in blue) gives the centre of the circle tangent to all three sides (known as the incircle).
The midpoint method finds the centre of mass of the triangle, assuming it to be a uniform lamina (eg, a piece of steel plating, or cut from plywood, etc). The lines here are known as medians.
The perpendiculars from corners method gives something called the orthocentre. Interestingly, the triangle made by joining the points at the foot of each perpendicular (known as altitudes) is the smallest perimeter triangle possible to be drawn within a triangle.
Uploaded by Anthony, TheChalkface.net