A border is a line round the edge of an element, or of each label. The same control sets the lines of a Line element, the rules of a Grid and the outline of a Text, so it's described once, here.

The border control
Size	How thick, in points. At 0 there's no border, and the rest is hidden.
Line style	Solid line, or one of the dashed and dotted patterns: Dash, Dot, Dash Dot, Long Dash, Tiny Dot and more.
Dash end style	For a dashed or dotted style: whether each dash has square (Flat) or Round ends.
Composition	One line (Single) or several side by side: Double thin, Double thick, Triple, Triple Inverted, Quadruple, Quadruple mixed.
Sketched style	Subtle, Medium or Heavy make the line wobble, as if drawn by hand. No sketch effect keeps it straight.
Antialias	Smooths the line's edges. It starts on.
Color style, Color, Color source	The color control: a color or a fade, or a color from your data. A border has no Transparent: set Size to 0 instead.

Where it is. An element's Border is in its Properties; it follows the element's shape. A label's is in the Label set panel and follows the shape of the labels. Either way it's drawn inside the edge, so a thicker border doesn't make anything bigger.

What the other uses add or leave out
Line	Rotation, Begin style and End style (Round, Arrow, Diamond and more), and Padding. Dash end style is always there.
Text outline	Outline style: drawn Behind the letters, in Front of them, or Empty - the outline alone, the letters hollow. No Antialias or Composition.
Grid	The size is called Line size, and the color is set apart.

The ⋮ menu next to Border has Copy border, Copy color, Paste and Reset all. A page element's border can't take its color from your data.