Is there any difference between $\mathbb{R}^3$ and Euclidean space denoted $\mathbb{E}^3$? I just can't seem to make any distinction between the two. If anyone has a simple explanation I'd like to hear it.
 A: $$(\mathbb{R}^3,d):=\mathbb{E}^3$$ where $d$ is the standard Euclidean metric. Here $\mathbb{E}$ denotes the "Euclidean"
Euclidean space is modeled by the real coordinate space $\mathbb{R}^n$ of the same dimension and it is denoted by $\mathbb{E}^n$ to emphasize on the Euclidean nature.  
$\mathbb{R}^3$ without any pre defined mathematical structure is just a coordinate space. 
If we define vector addition and scalar multiplication, it becomes a prototype of $n-$ dimensional vector space. If we define a topology $\tau$ on it $\mathbb{R}^3$ becomes a topological space. 
$\mathbb{E}^n$ is just used to denote the special case where $\mathbb{R}^n$ is equipped with Euclidean metric.
A: The set $\mathbb{R}^3 = \{(x_1,x_2x_3)\mid x_i \in \mathbb{R}\}$ is the set of real $3$-tuples. Very often, there is an extra structure on $\mathbb{R}^3$. For instance, $\mathbb{R}^3$ can be regarded as a vector space. Sometimes this extra structure is not explicitly specified, and it follows from the context what kind of space one means (vector space, metric space, ...). The Euclidean space is the set $\mathbb{R^3}$ with such an extra structure.
The Euclidian space $\mathbb{E}^3$ is the set $\mathbb{R}^3$ endowed with an Euclidean structure. Strictly speaking, we regard $\mathbb{R}^3$ as an affine space (often denoted with $\mathbb{A}^3$) and furthermore it is endowed with the Euclidean inproduct $\langle \cdot,\cdot \rangle$.
Firstly, the affine structure means that at every point $p \in \mathbb{E}^3$ there is a tangent space $T_p \mathbb{E}^3$ wich consist of all vectors $v_p$ (or $v$ in short) which have their "base point" at $p$. Without going in too much mathematical details, it means that we explicitly make a distinction between points $p$ (positions in space) and vectors $v$ (directions). For instance, an expression like $p+v$ (a point plus a vector) makes sense in this space. Very intuitively, it means that if we are in a point $p$ and walk in the direction of $v$, we arrive in a new point $q$.
The Euclidean space is the affine space endowed with the (usual) inproduct $\langle \cdot, \cdot \rangle$ at every point. That is, at every point $p$, the tangent space $T_p \mathbb{E}^3$ becomes an inproduct space. Confusingly enough, one often calls such an inproduct on every tangent space a  metric (as in Riemannian metric), but is it not a distance function as in topology, it is an inproduct. 
Using the inproduct at every tangent space, we can define the (usual) distance function on $\mathbb{E}^3$. If $p,q \in \mathbb{E}^3$, then $q-p$ can be regarded as a vector in $T_p \mathbb{E}^3$. We define the distance between two points $p,q$ as the length $\sqrt{\langle q-p,q-p\rangle}$ of the vector $q-p$ in $T_p \mathbb{E}^3$.
A: $\mathbb E^n$ is coordinate-free. $\mathbb R^n$ is (or can be identified with) $\mathbb E^n$ equipped with a rectangular coordinate system.
