What's the difference between interior and relative interior? As defined in Convex Optimization written by Stephen Boyd & Lieven Vandenberghe, both interior and relative interior seems to describe a same thing: a set that peels away its boundary points. So, what on earth is the difference between these two concepts ?
Here are the definitions of these two concepts from Convex Optimization:

interior: The set of all points interior to $C$ is called the interior of $C$


relative interior: its interior relative to the affine hull of $C$

 A: The relative interior of a set is the interior of the set when it is viewed as a subset of the affine space it spans.
For example, the interior of the segment connecting $(0,0)$ to $(1,1)$ in the plane is empty, but the relative interior is the open segment with those endpoints.
A: I came across this question whilst studying the same textbook, but I found that Mariano Suárez-Álvarez's gracious response, whilst, of course, appreciated, to be lacking the detail I desired. Fortunately, Justin Rising posted a more elaborate answer to the same question on Quora. I have decided to copy and paste it here for the benefit of others.
Justin Rising's response to the same question on Quora:

Consider $\mathbb{R}^3$ with the standard Euclidean metric, and look
at the set of points $(x, y, 0)$ such that $x^2 + y^2 \leq 1$.  What's
the interior of this set?
A point is in the interior of a set if you
can draw a small open ball around it which is itself contained in the
set.  But for any point in our set, any open ball around any point in
it will contain points outside the $x-y$ plane, and so it has empty
interior.
That's technically correct, but it's undesirable behavior.
The relative interior of a set is obtained by looking at the affine
hull (http://en.wikipedia.org/wiki/Affine_hull) of the set, which in
this case is $\mathbb{R}^2$, and using the appropriate metric there.
So while our set has empty interior, its relative interior is the
interior of the unit disc, which is what it "really should be".
Relative interiors are a big deal in convex optimization, where you'd
really be restricting the set of problems you can solve for no good
reason if you insisted on using interiors rather than relative
interiors.
It came up in the comments, and it's worth adding to the
answer, that there are very good reasons to talk about both interiors
and a relative interiors.  The relative interior is only defined in
vector spaces; in particular, you have notions of addition and scalar
multiplication to define an affine hull.  If I were to define a
topology on the words in the dictionary, I could talk about the
interior of a set immediately, but I'd have to do quite a bit of
legwork to be able to talk about relative interiors.

A: Put it simple and intuitive:
The relative interior of C is viewed from the affine space $\mathrm{aff(C)}$, and the interior of C is viewed from the whole space where C lies in, which is sometimes "bigger" than $\mathrm{aff(C)}$. E.g. in Boyd's book example 2.2: The relative of C is viewed from $\mathrm{aff(C)=R^2}$ while the interior of C is viewed from $\mathrm{R^3}$.
Inspired by the answer from Michal Adamaszek.
A: The interior $x$ of the set $C$ means that there exists a small ball around $x$,that is $\{y| \Vert y-x\Vert\leq r\}$  belongs to $C$. In this case, $y$ is a 3D ball while $C$ is a square plane, so  $C$ does not have interior.
The relative interior of the set $C$ is the intersection of the ball $y$ and aff $C$, so the relative interior is the set except the boundary.
