Are there different types of vectors? In linear algebra we have vectors:$$
\mathbf{A}=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z$$
We have vector algebra, i.e. vector addition, dot product, lines, planes, etc. A vector have a magnitude and a direction.
However, in multivariable calculus we also have vectors:$$
\mathbf{A}(t)=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z
$$
Here we do derivatives and integrals.
What is the difference?  Are there different types of vectors? 
I have always thought of vectors as the representation in the above link.

Let's start with vectors in linear algebra. I don't know how abstract your linear algebra class was, but in essence linear algebra focusses on the algebraic aspect of vectors: they live in a structure called a vector space and can be added, subtracted and scaled by numbers - that's all there is to it. This vector space can have any dimension: in your example the vector is in a three dimensional space, but lower or higher (even infinite) dimensions are also possible. Also the scalars (in your example the $x$, $y$ en $z$) can be any sort of numbers; I won't go into the abstract details, but for example you can let them be rational, real or complex numbers and this choice has minor impact on how the algebra works out.
What's important is that multivariate calculus relies on a more specific sort of vector, since two restrictions are made: the scalars must be real numbers and the dimension must be finite. These vectors are sometimes called euclidean and are what your linked Wikipedia article is about. In this specific case, as you mentioned, vectors can have an analytic structure as well and you can describe and study things like continuity, derivatives and integrals with these vectors.
TLDR; multivariate calculus vectors are a specific instance of linear algebra vectors: they are scaled by real numbers and live in a vector space of finite dimension, often denoted as $\mathbb{R}^n$ where $n$ is the dimension.
