A simple vector having dimension? I am aware that a matrix has a dimension
    [1  2  3  1]
A=  [3  3  4  6]
    [0  2  2  3]

Here we say that a matrix, in this case A, has the dimensions of 3 by 4. 
What about vectors? 
Do vector have a dimension?
Consider the following:
u = [1 2 3]

Is this u vector? Does it have a dimension? (1 by 3?)
x = [1]
    [2]
    [3]

Is this x a vector? Does it have a dimension? (3 by 1?)
Also if I have a lowercase letter with a column or row of one, we say it is a vector?
(Basically, I am lost on this idea who has this idea of dimension, how to we safely say something is a vector but please also answer the question above because I am more of a example based person)
 A: They could be vectors.
An object by itself is not a vector. It is a vector if and only if it is part of a vector space. And a vector space is a collection of objects which one can add together, and scale, subject to a certain list of rules.
The set of all $3\times 1$ matrices with real entries, using standard matrix addition and scaling by real numbers, is a (real) vector space. Your $x$ is an element of that vector space, and as such a vector. The set of all $3\times1$ matrices with no negative entries, again using standard addition and scaling, is not a vector space (you can't scale those by $-1$, so that set of matrices violates the rules). $x$ is also an element of that set, and as such not a vector.
The set of all matrices of any given, fixed dimension with real entries, using standard matrix addition and scaling by real numbers, will always be a vector space. Mix in restrictions, and you have your typical textbook exercise (variations on "Is the set of such-and-such matrices a vector space?" is really common, even on this site).
Now, usually, when talking about vectors in the context of matrices, one specifically means either column vectors, which are $m\times 1$ matrices like your $x$, or row vectors, which are $1\times n$ matrices like your $u$. Thinking of them as special cases of matrices, with all that includes of matrix addition, scaling, and matrix multiplication is not at all problematic. In fact, this is the entire reason matrices are as popular as they are and work as well as they do.
