Is there a definition of "direction" in physics? Is there an actual definition of "direction" (that is, spatial direction) in physics, or is it just one of those terms that's left undefined? In physics textbooks it's always just taken for granted that the reader knows what it means (and it is true that just about everyone does indeed have an "intuitive" idea of what it is). But it would be more satisfying to have an concrete definition, if possible.
 A: In a slightly philosophical vein, direction acquires meaning only when you compare two objects. For example, when you attribute a "direction" to a vector,you are comparing it to the basis set of that vector space. By comparing I mean taking the inner product. I must emphasize that "direction" has meaning only as a representation.
A representation is a list of coefficients from the field over which the vector space is defined.
A representation is meaningless unless the basis set is specified. If somebody gives you a list of three numbers (a,b,c), there is not much you can say unless they give you the basis set $\{v_{i}\}_{i=1,3}$
A: I'll take this to a different level of abstraction, since you seem to want a more Philosophical approach.
Given a measure of distance between two places $x$ and $y$, $d(x,y)$, a concept of direction can be formed by considering all the dimensionless ratios of distances between pairs of places from a set $P=\{x_1,x_2,...,x_n\}$. In some cases we might convert a dimensionless ratio $-1\le r\le 1$ into an "angle" $\arccos(r)$ (which is a purely algebraic assignment that will only be helpful in some cases). If there is a (not necessarily unique) subset $S$ of the dimensionless ratios of distances that determines all the other dimensionless ratios, we might consider those to be "directions", relative to that subset. If I give you the directions $S$ and the way in which those directions determine the whole set of dimensionless ratios, and nothing extra, you can reconstruct the whole set of dimensionless ratios $R(S)$, or, more interestingly, if I give you a way in which directions determine the whole set of dimensionless ratios and give you a subset $U\subset S$, that determines $R(U)$, which is, so to speak, where I want you to go. [There are many other mathematical constructions one could contemplate, but space here and my time and everyone else's are all finite.]
This depends on us being able to identify $n$ "places" and on us assigning as many as $\frac{n(n-1)}{2}$ distances between them. How one does that in elementary cases is learned in geometry classes, at increasingly sophisticated levels from before kindergarten to beyond Euclid. At some point there is a move from a finite number of points/places to a countably infinite number of points/places, and then, if one doesn't mind such things, to an uncountably infinite number of points/places, but those are just ways to fill in the between (I feel happier filling it up, giving a space a continuous topology, but that's a prejudice and it's not clear that it's necessary for Physics).
It's not uncommon in Mathematical Physics to assign distances to pairs of much more abstract places. I work with individual functions or with sets of individual functions as a single place. Where differences arise, it is often because the set of places cannot be embedded in a low-dimensional space. In general it will be possible to embed a set of places and the distances between them into a lower-dimensional space than otherwise if one allows the space to be curved. Another significant difference arises if there are negative as well as positive distances, which prevents whatever system one constructs being embedded into a Euclidean space of any dimension, but might allow it to be embedded into a Minkowski or pseudo-Riemannian space. 
A Physicist's criteria for how useful any given such assignment of places and distances might be will presumably include some notion of reproducibility, which gets into more difficult abstract territory, including the now ever-present relationship between probability and statistics.
Another Question, of course, is "What is Distance"? Ultimately I suppose one comes to some irreducible definitions, where one throws up one's hands and asks whether we're just going to talk about stuff or do something interesting with what we all just know, until someone points out persuasively that we don't and that it's useful to just know something different.
A: I reckon it describes a non-commutative relationship between two objects (or events) which is completely determined by their position in some (measurable) space, independent of their distance (by some well defined measure of distance) and some set of operations on both of them (probably just translation really).
Can anyone do better? (I expect they can)
As an aside, I guess you could use a method more suited to analytical philosophy. If I wanted to examine what a direction as used by physicists was, I would do something like:
1) Acknowledge that most physicists would agree on what a direction is.
2) Look at how they use it and write about it, I expect I would see that it is a relationship between two things, that it requires some concept of a space, probably some set of coordinates etc,
3) Try and concisely write down what I learned, probably tearing my hair out at the same time.
4) go to 2 and check what I wrote
A: Take the tangent vector at the start of the shortest path between a pair of distinct points, and mod out by distance.  What you have left is "direction".  In a Euclidean space with a standard metric, you can do this by just dividing any non-zero vector by its length.  The resulting normalized vector is a "direction".  For curved spaces, it is a bit more complicated, but still the same basic idea.
A: The definition of direction is by three numbers--- (a,b,c) which represent the components of a vector along an arbitrary x-axis, y-axis, and z-axis. But if you ignore the motivation, a vector is a triplet of numbers. This reduces the description of direction to the description of real numbers, and has been the standard definition since the time of Descartes.
The reason this definition took so long to formulate is because there is an arbitrariness in the choice of coordinates. Given a collection of directions (triplets of numbers) which describe a physical situation, one can always transform all the directions by a rotation matrix (which is defined noncircularly as a matrix all of whose columns are perpendicular to each other and have unit length), and the situation is not altered.
Because of the rotation business, mathematicians are often uncomforable with the definition of direction by triplets of numbers, but this is really the best way, since all other methods are more complicated and needlessly so. The issue of an arbitrary coordinate frame to describe invariant objects is important for other reasons, and is central to gauge theories.
LATER EDIT: The proper definition of a pure direction is the ratio of a nonzero vector to its length, which is algebraically simple, because the length is the square root of the sum of the squares of the components. This defines a pure direction. But a vector with direction and magnitude is simply the three numbers together. This obvious clarification is because of a downvote and comment below.
A: Simply: 
Points are well defined in a space (e.g Euclidean space). Now consider two points. One can assign two information to these two points 1) distance  and 2) direction. I skip the concept of the distance. The choice (or information)of which point be first and the other be second point defines the direction.
A: Man, some of the answers are way of the heads of morons like me, but anyway...
At a basic level, a vector is construced from the difference between two coordinates a-b. The number that represents the distance between them is the magnitude of displacement vector ab. There is nothing else, certainly no direction intrinsic to it. Introducing a third coordinate means you can define a second displacement vector ac. Let's suppose the magnitude of ab = ac. We can define the angle between ab and ac as the ratio of the arc distance between b and c to the circumference of a circle of radius equal to the magnitude of ab=ac. That's one way of doing it, but essentially you're just assigning a number to a pair of vectors.
Direction is measured by angle and therefore is a function that assigns a number to two vectors. It isn't intrinsic to a vector.
A: There is no reference to “direction” of a vector in the definition of a vector or vector space. However, we may have various examples of vector spaces. For example, the real numbers form a vector space ( without any direction). As another example, geometric arrows can form a vector space. In latter example, arrows are sketched with “direction” respect to a reference (implicitly or explicitly) for example with respect to edge of the paper or an axis or an oriented line connecting two points.
