What is the meaning of $||x||=\sqrt{\langle x,x\rangle}$ I understand that a norm assigns a length to each vector in a vector space. 
I have been told that $$||x||=\sqrt{\langle x,x\rangle}$$ is a norm. So does this equation find the length of vector $x$, and what do the $x$'s under the square root imply? 
Are they just the start and end points of the vector $x$? 
 A: The notation
$$\langle\cdot,\cdot\rangle$$
is used to denote the inner product, which in Euclidean space is essentially the dot product. Using the dot product definition in the case of the inner product of a vector $\mathbf{v}$ and itself,
$$\mathbf{v}\cdot\mathbf{v}=v_x^2+v_y^2+\cdots$$
Notice that this is simply the magnitude of $\mathbf{v}$ squared, because of the Pythagorean theorem:
$$\mathbf{v}\cdot\mathbf{v}=\langle v,v\rangle\to\sqrt{\langle v,v\rangle}=\sqrt{\mathbf{v}\cdot\mathbf{v}}=||\mathbf{v}||$$

For two vectors $\mathbf{a}$ and $\mathbf{b}$, the dot dot product is simply the product of their magnitudes multiplied by $\cos\theta$, where $\theta$ is the angle between the two vectors. When $\mathbf{a}=\mathbf{b}$, $\theta=0$, and so the result is simple the magnitude of the vector squared - which is typically represented as the "length" of the vector squared.

Are they just the start and end points of the vector x?

No. They refer to the magnitude of the vector $\mathbf{x}$.
Here's a geometric representation that Piwi suggested:

A: In your case $\langle x,x\rangle$ is the scalar product of the vector $x$ with itself.
A: You can see a vector in $\mathbb{R}^n$ as an oriented segment in $n$-dimensional space that start from the origin and goes to the point $X$ of coordinates $(x_1,x_2,\cdots,x_n)$.  So the inner product $\langle x,x\rangle$ is :
$$
\langle x,x\rangle= x_1^2+x_2^2+\cdots+x_n^2
$$
and $\sqrt{\langle x,x\rangle}$ is the usual Pythagorean lenght of the segment from the origin to $X$.
This is the geometric intuition behind the concept of norm that is extend, in this form linked to an inner product, to any space that has an inner product. 
A: You probably mean the inner product $\langle \boldsymbol{x}, \boldsymbol x \rangle$.
If you have two vectors $\boldsymbol a = (a_1,a_2,\dots ,a_n)$ and $\boldsymbol{b} = (b_1,b_2,\dots,b_n)$, then the inner product $\langle \boldsymbol{a},\boldsymbol{b}\rangle$ is defined as $$\langle \boldsymbol{a},\boldsymbol{b}\rangle = \sum^n_{i=1}a_ib_i = a_1b_1 + a_2b_2 + \dots + a_nb_n.$$
Note that $\langle \boldsymbol x, \boldsymbol x\rangle = \sum^n_{i=1} x_i^2 = x_1^2 + x_2^2 + \dots +x_n^2$. If you take the square root  of $\langle \boldsymbol x , \boldsymbol x\rangle$ you obtain the length of the vector $\boldsymbol x$ from the origin (notice the resemblance with the Pythagorean theorem).
