Thousands of years ago, when the Greek philosophers were laying the first foundations of geometry, someone was experimenting with triangles.
They bisected two of the angles and noticed that the
angle bisectors crossed.
They drew the third bisector and surprised to find that it too went through the same point. They must have thought
this was just a coincidence.
But when they drew any triangle they discovered that the
angle bisectorsalways intersect at a single point!
This must be the 'center' of the triangle. Or so they thought.

After some experimenting they found other surprising things. For example the
altitudes
of a triangle also pass through a single point (the orthocenter).
But not the same point as before. Another center! Then they found that the
medians pass through yet another single point.
Unlike, say a circle, the triangle obviously has more than one 'center'.

The points where these various lines cross are called the triangle's
points of concurrency.

Some triangle centers

There are many types of triangle centers. Below are four common ones.
There is a page for each one. Click on the link to probe deeper.

In the case of an equilateral
triangle, the incenter, circumcenter and centroid all occur at the same point.

How many centers does a triangle have?

Lots. Over time mathematicians have found many more. Some with names like 'Apollonius Point' and 'Symmedian point'.
Many are exotic and beyond the scope of this volume, but
you might find this huge list interesting:
http://mathworld.wolfram.com/topics/TriangleCenters.html