In What Order Should I Send My Vertices To OpenGL for Culling

I learned another rule of thumb (literally) for determining the vertex order known as the "right hand rule".
Imagine your open hand (right) inside the cube with your thumb pointing towards the center of the cube. If you then curl your hand into a fist, your fingers will pass the vertices in the correct order. Since you are using your right hand for this it is called the "right hand rule".

Conversely, if you start with your left hand and point your thumb away from the center of the cube, your fingers will again sweep the vertices in the correct order. This is known as the "left hand rule" (surprise).

Both methods work to give you counter-clockwise ordering. For clockwise ordering, just use the opposite hand.

By default? In counter-clockwise order.

Consider a triangle facing the camera:

A
|\
| \
|  \
B---C

A->B->C would be front facing (counter-clockwise order), A->C->B would be rear-facing (clockwise order).

You can change which way OpenGL considers "front facing" via glFrontFace():

The projection of a polygon to window coordinates is said to have clockwise winding if an imaginary object following the path from its first vertex, its second vertex, and so on, to its last vertex, and finally back to its first vertex, moves in a clockwise direction about the interior of the polygon. The polygon's winding is said to be counterclockwise if the imaginary object following the same path moves in a counterclockwise direction about the interior of the polygon. glFrontFace specifies whether polygons with clockwise winding in window coordinates, or counterclockwise winding in window coordinates, are taken to be front-facing. Passing GL_CCW to mode selects counterclockwise polygons as front-facing; GL_CW selects clockwise polygons as front-facing.

By default, counterclockwise polygons are taken to be front-facing.

For ordering your vertices, consider an ideal cube:

  6---7
 /|  /|
2---3 |
| 4-|-5
|/  |/ 
0---1

For each face mentally rotate it to face the camera (your mind's eye):

Sides:
2---3  3---7  7---6  6---2
|   |  |   |  |   |  |   |
|   |  |   |  |   |  |   |
0---1  1---5  5---4  4---0

Bottom/Top
0---1  6---7
|   |  |   |
|   |  |   |
4---5  2---3

Then you can just visually read off the quads or triangle pairs in the right counter-clockwise order:

2---3                3         2---3 
|   |  becomes      /|   and   |  / 
|   |             /  |         |/ 
0---1            0---1         0 

Triangles 0-1-3 and 0-3-2
Quad 0-1-3-2

It doesn't matter which vertex you start the triangle/quad with, for example with the first triangle 0-1-3, 1-3-0, and 3-0-1 are all equally valid, front-facing triangles.