What is a vector? What is a vector?
As the question says what is a vector and what are its uses or, I mean, when should we use vectors?
Is this a branch of geometry or algebra or trigonometry?
 A: The basic geometric vectors are measurements which have a non-negative magnitude and a geometric direction, with the caveat that we treat all vectors with zero magnitude as the same (the zero vector.)
The most obvious example is, given two point, $A$ and $B$ in (Euclidean) space:  what direction would I have to travel to get from $A$ to $B$ in a straight line, and how far would I have to travel?
Other examples are things like the velocity of an object, or the object's momentum.
Vectors are interesting because we can "add" them (when the units are the same, as with all quantities.) If we take the "travelling" vector above, then if we travel in the direction of vector $v$ for the distance equal to the magnitude of $v$, and then travel in the direction of $w$ for the distance of $w$, we get a new vector $v+w$.  It turns out that $v+w=w+v$, which is convenient. 
(Momentum is interesting, because if two objects collide and combine, one with momentum $v$ and the other with momentum $w$, it turns out the new object will have momentum $v+w$. So it is not necessary to think of vector addition as necessarily "follow one vector then the other" - rather, it is a combination. Even if the objects don't combine, but rather, glance off each other, we still have a nice property. If A had momentum $v_1$ before they collided and $v_2$ after, and $B$ had momentum $w_1$ before and $w_2$ after, it turns out, by "conservation of momentum," that $v_1+w_1=v_2+w_2$.)
You can also scale a vector by a constant, so if $r$ is any positive number, you create a vector $rv$ which is the same direction as $v$ buth with the magnitude equal to the magnitude of $v$ times $r$. There is also a notion of $-v$, which is the vector with the opposite direction to $v$ and the same magnitude, and $v+(-v)=0$, the zero vector.
These are the first types of vectors that you encounter, but the field of linear algebra introduces whole new, more abstract notions of vectors that are also quite useful. Essentially, mathematicians noticed a lot of cases where you had a collection of "vectors" $V$ and a "field of numbers" $F$ where you can add any two vectors, and you can scale any vector by any element of $F$. There are some properties required on addition and scalar multiplication to make this a vector space.  In these cases, the vectors in the vector space  might not be viewable as "direction and distance."  There is no notion of the angle between two vectors, either. We still think of these general vector spaces as essentially "geometric," but in a far more abstract way.
A: First: what is a vector?
Given a vector space $V$ over for example the real numbers $\mathbb{R}$. Say that the dimension of the vector space is $n$. Then a vector is a $n$-tuple of numbers. For example if $n = 2$ and $V = \mathbb{R}^2$, then 
$$
(1,2)\quad\quad (0,0)\quad\quad (-2,7)
$$
are examples of vectors.
Now this almost makes is sound like a vector is just a point, and in a certain sense it is just a point. However, where we can't technically add points, we can add vectors. We add two vectors by adding their coordinates. For example
$$
(1,2) + (-2,7) = (1 + (-2) , 2 + 7) = (-1, 9).
$$
We can also for example multiply a vector by a real number, and we do that by multiplying each coordinate by that number. Examples:
$$
3(2,7) = (3\cdot2, 3\cdot 7) = (6, 21)\\
0(2,7) = (0\cdot 2, 0\cdot 7) = (0,0).
$$
To get a complete understanding of what a vector is, you might want to look up the definition of a vector space because the axioms of a vector space is exactly what describes what you can do with vectors.
The above then is just one description of a vector. There are other ways to define them.
Second: Maybe you real question is how we should think about a vector. From the above you can think about a vector, for example $(2,7)$ as an arrow that starts at the origin of a coordinate system and ends at the point $(2,7)$. That way we can think about a vector as 


*

*a direction

*and a magnitude (being the length of the arrow)


That being said, of you start at any other point and you draw an arrow with the same direction and magnitude, then the two arrows are the same when considered representatives of the vector.
Third: What are vectors good for? Vectors are very helpful in many situations. One example is when you consider a physical object with a force working on it. By that I mean for example a football that is pulled towards Earth by the force of gravity. The (sum of the) forces that work on an object determine how the object moves. And it turns out that vectors are very helpful in thinking/working with forces. So a force is indeed something that has a direction (i.e. something being pulled in a certain direction) and a magnitude (i.e. how hard is the thing being pulled).
A: A vector is an element of an Vectorspace. It's a branch of linear algebra, there is a lot to tell about vectors, where did you met a vector?
Well a vector can be nearly anything, for example every element of a field is a vector. You can use vectors to describe positions, if you take an element of $\mathbb{R}^3$  $(x,y,z)^T$ you are able to describe a position by telling it coordinates.
A: Another kind of motivation, which was given eg by Richard Feynman (Lectures on Physics) for a vector in three real dimensions is that a vector is a set of three numbers which transforms in a certain way under rotations. The formula for the rotation of a vector is compatible with the (supposed) Euclidean geometry of space, and this compatibility is such that it enables us to write the equations of physics in a manner independent of arbitrary translations and rotations (the choice of a basis).
So, continues Feynman, if we can write a physical law in vector form we know that it is compatible with the basic conservation and invariance principles of physics.
So in a sense vectors are the natural language for Newtonian physics in three dimensional Euclidean space.
And from that starting point the right way of writing General Relativity in non-Euclidean space-time grows.
And in another development pure mathematicians discovered uses for abstract vector spaces (eg in describing extensions of fields) which go beyond the physical ideas of magnitude and direction.
