The dot product gives a number as output, and it's an operation that comes up naturally when you study the angle between two vectors: the angle $\theta$ between vectors $\mathbf{v}$ and $\mathbf{u}$ is given by $\displaystyle \cos \theta = \frac{\mathbf{v}\cdot \mathbf{u}}{\|\mathbf{v}\|\|\mathbf{u}\| }$.
This product makes sense in any number of dimensions. So for example, in 4 dimensions, we could still define the angle between two vectors such as $(2,-1,3,6)$ and $(4,0,-3,7)$, even though we cannot visualize them...
The cross product on the other hand is only defined in three dimensions, and gives a vector as answer. So it is quite a different kind of thing. It comes up when you want to go from 2 to 3 dimensions in the following way: if you start with two 2-dimensional vectors such as $\mathbf{v}=(3,4)$ and $\mathbf{u}=(-1,3)$, we can think of them as actually living in 3-space by adding a $0$ as third coordinate: $\mathbf{v}=(3,4,0)$ and $\mathbf{u}=(-1,3,0)$. Then if we take the cross product $\mathbf{v}\times \mathbf{u}$, the result will be a vector perpendicular to both of them, so it will point in the third dimension, the $z$-axis...
