Try this
Adjust the polygon below by dragging any orange dot. Note that the length of all sides remains the same and the
interior angles remain the same.

Regular polygons tend to look like someone tried to make a circle out of some straight lines.
In fact, if you have a polygon with very many sides, it looks a lot like a circle from a distance.
The sides and vertices are evenly spread around a central point, and regular polygons are
convex
- all the vertices point 'outwards'.

In the figure above, you can change the number of sides using the controls on the right of the polygon.
An interesting thing to note is that once the number of sides is quite high (say more than 17) the shape begins to look a lot like a circle.
In fact, as you keep adding sides, the area of the polygon,
the area of its incircle,
and the area of its circumcircle all converge on the same value.

Attributes

Vertex

The vertex (plural: vertices)
is a corner of the polygon. In any polygon, the number of sides and vertices are always equal.

Side

The sides are the straight line segments that make up the polygon. See Sides of a Regular Polygon
for more information and formulas used to calculate their length.

Apothem (inradius)

The apothem of a regular polygon is the line from the center to the midpoint of a side. (or, the length of that line). It is
also the radius of the incircle. See Apothem of a Regular Polygon

Radius (circumradius)

The radius is the distance from the center to any vertex. It is also the radius of the polygon's circumcircle,
the circle that passes through every vertex. See Circumcircle of a regular polygon

By definition, all sides are the same length, so the perimeter is simply the length of a side times the number of sides.

All vertices lie on a circle

In a regular polygon all its vertices lie on a circle. In fact, this can be one of the definitions of a regular polygon:
"All sides are the same length and all vertices lie on a circle". This circle is also called the circumcircle. See
Circumcircle of a regular polygon

Being equilateral is not enough

For a polygon to be 'regular' it must have all sides the same length and all interior angles the same.
The figure on the right is actually an example of an
equilateral polygon since it has all sides the same length,
but it is not a regular polygon because its interior angles are not all the same.

Also, this violates the rule that in a regular polygon, all vertices must lie on a circle.
See Irregular Polygon.