Here is how I think of it, lets construct a vector from scratch using just two points in space A,B. Draw a line segment between the two points A,B. The magnitude of the line segment is the 'length' of the vector. The 'orientation' of the line segment we can define as the angle that the line segment makes with the horizontal axis. To be clear this angle is measured counterclockwise from the positive x axis and is an angle between 0 and 180. OK so far we just have a line segment that is situated on the plane somewhere and we know how it is oriented, but there is no arrowhead yet. Now the last thing you need is the 'sense', that basically tells you where to put the arrowhead and implies an order. We can define this vector as AB with an arrow over it, where you read it as the vector starting from A and ending at B.
More generally you can define a vector by defining its magnitude (length), its orientation, and then its sense. But keeping in mind, technically a vector is an equivalence class. So there are an infinite number of vectors which are parallel to each other (have the same orientation) and have the same sense or same choice of where to put the arrowhead (there are only two possible senses, since the arrowhead can be placed on A or on B). But I don't want to confuse you. Usually we discuss vectors that are situated at the origin so we don't concern ourselves over other equivalent vectors.
I just checked wiki which is basically the same as what I wrote: http://en.wikibooks.org/wiki/Statics/Forces_As_Vectors#Graphically
"A vector may be represented graphically by an arrow. The magnitude of the vector corresponds to the length of the arrow, and the direction of the vector corresponds to the angle between the arrow and a coordinate axis. The sense of the direction is indicated by the arrowhead."
This pretty much sums up what I just wrote above. You see the 'direction' (your article uses the word orientation) just gives you an angle that the vector makes with the horizontal axis, but that creates an ambiguity since an arrow can point in two opposite directions and still make the same angle. The sense clears this ambiguity and indicates where the arrowhead actually goes. So the sense tells us the order so to speak in which to read the vector. It indicates where to start and end on the vector.
(Also technically you can indicate the angle of the vector however you choose, doesn't have to be a horizontal axis, could be vertical).
Now you are probably more familiar with a vector description that combines orientation and sense by saying for example, give me the vector that is 2 units length and is rotated 30 degrees counterclockwise starting from the point (2,0). But I am combining direction and sense here. Technically I could have said, the vector is 2 units length (magnitude), the vector makes an acute angle of 30 degrees with the horizontal axis (orientation/direction), and the starting point of the vector is (0,0) and ends at (2 cos 30, 2 sin 30) this is 'sense'.
and together, orientation and sense determine 'direction'
In math you just have to be flexible. Different authors mean the same thing but use different words. In an ideal world every author would agree with each other's notation and terminology. Until we reach a universal mathematical language, it is better to try to get an understanding.