So, your first step is indeed correct, but the one you are not sure about is wrong. You are probably confused by your own notation, I prefer to write $\partial_i=\frac{\partial}{\partial x_i}$ instead of $\partial x_i$. And remember that repeated indices are being summed over, so e.g. $u_ju_j=\mathbf{u}\cdot\mathbf{u}$. Given that, you have
\begin{equation}
u_j\partial _i u_j-u_j\partial_j u_i
=(1/2)\partial_i(u_j u_j)-(\mathbf{u}\cdot \nabla)u_i
=(1/2) \partial_i (\mathbf{u}\cdot\mathbf{u})-(\mathbf{u}\cdot \nabla)u_i,
\end{equation}
which (taking into account your earlier work) is the $i$-th component of the equation
\begin{equation}
\mathbf{u}\times(\nabla\times \mathbf{u})=\frac{1}{2}\nabla(\mathbf{u}\cdot\mathbf{u})-(\mathbf{u}\cdot \nabla)\mathbf{u}
\end{equation}
A scalar is a number, like 3 or 0.227. It has a bigness (3 is bigger than 0.227) but not a direction. Or not much of one; negative numbers go in the opposite direction from positive numbers, but that's all. Numbers don't go north or east or northeast. There is no such thing as a north 3 or an east 3.
A vector is a special kind of complicated number that has a bigness and a direction. A vector like $(1,0)$ has bigness 1 and points east. The vector $(0,1)$ has the same bigness but points north. The vector $(0,2)$ also points north, but is twice as big as $(0,2)$. The vector $(1,1)$ points northeast, and has a bigness of $\sqrt2$, so it's bigger than $(0,1)$ but smaller than $(0,2)$.
For directions in three dimensions, we have vectors with three components. $(1,0,0)$ points east. $(0,1,0)$ points north. $(0,0,1)$ points straight up.
A scalar field means we take some space, say a plane, and measure some scalar value at each point. Say we have a big flat pan of shallow water sitting on the stove. If the water is shallow enough we can pretend that it is two-dimensional. Each point in the water has a temperature; the water over the stove flame is hotter than the water at the edges. But temperatures have no direction. There's no such thing as a north or an east temperature. The temperature is a scalar field: for each point in the water there is a temperature, which is a scalar, which says how hot the water is at that point.
A vector field means we take some space, say a plane, and measure some vector value at each point. Take the pan of water off the stove and give it a stir. Some of the water is moving fast, some slow, but this does not tell the whole story, because some of the water is moving north, some is moving east, some is moving northeast or other directions. Movement north and movement west could have the same speed, but the movement is not the same, because it is in different directions. To understand the water flow you need to know the speed at each point, but also the direction that the water at that point is moving. Speed in a direction is called a "velocity", and the velocity of the swirling water at each point is an example of a vector field.
I think the only other thing to know is that in one dimension, say if you had water on a long narrow pipe instead of a flat dish, vectors and scalars are the same thing, because in one dimension there is only one way to go, forwards. Or you can go backwards, which is just like going forwards a negative amount. But there is no north or east or northeast. So one-dimensional vectors are interchangeable with scalars: all the vector stuff works for scalars, if you pretend that the scalars are one-dimensional vectors.
If this isn't clear please leave a comment.
Best Answer
It's important to note that in the first, $v$ is a vector function while in the second, $v$ is a scalar function. Try writing out each bracket first: $$\nabla v = \partial_i v\quad \text{so}\quad u \cdot (\nabla v) = u_i \partial_i v$$ while
$$u \cdot \nabla = u_i \partial_i \quad \text{so} \quad [(u \cdot \nabla) v]_j=u_i\partial_iv_j.$$