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