In the case of a triangle, there is always an incircle possible, no matter what shape the triangle is. In the figure above, the red circle is the incircle of the triangle.
It is possible to construct the incircle with a compass and straightedge. See Constructing the Incircle and Incenter of a Triangle
Regular polygons, (polygons that have all sides the same length and all interior angles congruent) can have incircles. As with the triangles case, each side of the polygon is a tangent to the incircle.
The center of the incircle, the incenter, is also considered to be the center of the polygon itself, since it is equidistant from each vertex.
For more on this see Incircle of a Regular Polygon and Regular Polygon definition.
Irregular polygons are not thought of as having an incircle or even a center. If you were to draw a polygon at random, it is unlikely that there is a circle that has every side as a tangent. An exception is the 3-sided polygon (triangle). All triangles always have an incircle (see above).