Why can vector components not be resolved by Laws of Vector Addition? A vector at any angle can be thought of as resultant of two vector components (namely sin and cos). 
But a vector can also be thought of resultant or sum of two vectors following Triangle Law of Addition or Parallelogram Law of Addition, as a vector in reality could be the sum of two vectors which are NOT 90°.The only difference here will be that it is not necessary that components will be at right angle.
In other words why do we take components as perpendicular to each other and not any other angle (using Triangle Law and Parallelogram Law).
 A: Vadim's answer resolves the Q very nicely. I would just like to add that given a set of n vectors in a vector space you can always construct a set of n orthonormal vectors which can then be used as a basis. This is well known as the Gram–Schmidt  orthogonalization process. 
A: Suppose you are attempting to find a point on a map given a starting point. Which do you prefer:


*

*4 km north and 3 km east of your current location

*7 km north and 4.2 km southeast of your current location


We can use whatever non-parallel vectors we want to describe an offset in two dimensions. Is it now clear why north and east are preferable to north and southeast?
A: Indeed, any vector can be resolved in terms of two components (in $n$-dimensional space in terms of $n$ components). For this being possible the components should be linearly independent, i.e. in your case they should not be parallel. The advantage of using two orthogonal/perpendicular components is that their scalar product is zero, which simplifies the math when calculating the coefficients:
\begin{array}
\mathbf{F} = F_x\mathbf{e}_x + F_y\mathbf{e}_y \Longrightarrow F_x = \mathbf{e}_x\cdot \mathbf{F}, F_y = \mathbf{e}_y\cdot \mathbf{F}.
\end{array}
Indeed,
\begin{equation}
\mathbf{e}_x\cdot \mathbf{F} = \mathbf{e}_x (F_x\mathbf{e}_x + F_y\mathbf{e}_y) = F_x\mathbf{e}_x \cdot\mathbf{e}_x + F_y\mathbf{e}_x \cdot\mathbf{e}_y = F_x,
\end{equation}
and similarly for $F_y$, since
\begin{equation}
\mathbf{e}_x \cdot\mathbf{e}_x = 1, \mathbf{e}_y \cdot\mathbf{e}_y = 1, \mathbf{e}_x \cdot\mathbf{e}_y = 0.
\end{equation}
For non-orthogal vectors $\mathbf{e}_x \cdot\mathbf{e}_y \neq 0$, the math becomes a bit more complicated and the interpretation of the projections as coordinates is less intuitive.
There are however some cases where using non-orthogonal components is beneficial, notably when dealing with non-orthogonal crystal lattices, such that of graphene (a hexagonal lattice.)
A: There are two main cases of separating into components: standard coordinates system, and separating into normal and parallel.
In a standard coordinate system, there is a set of coordinates, and points are given in terms of those coordinates. Not all coordinate systems are vector spaces; for instance, the longitude latitude system isn't, since the Earths surface is curved, and in general relativity, the coordinates are only locally vectors, since space-time is curved. But for flat space, we can treat the coordinate system as consisting of an origin, each point being represented by a vector pointing from the vector to that point, and the coordinates being given by the projections of that vector onto basis vectors. Generally, orthogonal basis vectors are chosen because that makes things simpler.
The other case where a vector is separated into components is where there is some reference vector, and other vectors are separated into normal and parallel to that. The most common reference vector is the gravitational vector: the parallel direction is treated as a distinguished axis of up/down, and other directions are sideways. But there are other examples of reference vectors: for instance, if you're pushing a cart up a slope, you might take the slope as a reference vector. Then if you have another vector, such as the force on the cart, you can resolve it into a component parallel to the slope, and another perpendicular to the slope. And of course if one vector is parallel to the reference vector, and another is perpendicular, then they will be perpendicular to each other.
A: This is mainly because the X and Y axes are also perpendicular to each other and therefore to measure quantities along these axes the vectors are resolved into X and Y components.
A: The vectors can be arbitrary and there are good reasons explained in the other answers. But I would like to approach this form another angle.
Often you as the person doing the problem setup can choose whatever coordinate system you like. Then, unless there is a apparent benefit of not choosing a orthogonal basis, you would probably choose a orthogonal basis for expediency and ease of decomposing of vectors. In the same way as you would like to prefer choosing the length of your all your basis vector to be 1. Again you could choose different values for different axes, but it makes reasoning about your results easier if you chose easily understandable bases. Again unless you have a reason to do otherwise.
Changing between bases is easy enough.
A: Because they have nothing to do with vectors and their addition. They are simple sketches or representations of algebraic objects, elements of vetoor spaces.
