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 $C$
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.
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.
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.
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.
This is a rephrasing of Thoerem 13 of Chapter 2 in Real Mathematical Analysis by Charles Pugh.
Theorem (Inheritance Principle). Every metric subspace $X$ of metric space $Y$ inherits its topology from $Y$. More precisely, every subset $U \subseteq X$ is open in $X$ if and only if there is a subset $V \subseteq Y$ open in $Y$ such that $U = V \cap X$.
This shows why the relative interior of $U$ is defined as
$$\mathrm{ri}(U) = \{x \in U: \exists \epsilon > 0, B_{\epsilon}(x) \cap \mathrm{aff}(U) \subseteq U\},$$
which is the set of points for which there exists an open ball in the background metric space such that its intersection with the affine hull containing $U$ lies inside $U$. Compare this to the definition of interior of $U$ which reads as
$$\mathrm{int}(U) = \{x \in U: \exists \epsilon > 0, B_{\epsilon}(x) \subseteq U\}.$$
Here, $\mathrm{aff}(U)$ plays the role of $X$ and $B_{\epsilon}(x)$ is an open ball in $Y = \mathbb{R}^n$. The theorem tells us that $B_{\epsilon}(x) \cap \mathrm{aff}(U)$ is open in $\mathrm{aff}(U)$, which are the open neighborhoods we really want for determining interior points of $U$.
Example. Take $U = \{(x_1, x_2, 1) \in \mathbb{R}^3: |x_1| \leq a, |x_2| \leq b\}$ with $a, b > 0$ and $Y = \mathbb{R}^3$. Here, the affine hull containing $U$ is $\mathrm{aff}(U) = \{(x_1, x_2, 1): x_1 \in \mathbb{R}, x_2\in\mathbb{R}\}$. Clearly, $U \subseteq \mathrm{aff}(U)$ but it is not open in $\mathbb{R}^3$ since no three dimensional ball can be contained in $U$ and $\mathrm{int}(U) = \varnothing$. Now, let $B_{\epsilon}((x_1, x_2, 1)) = \{(x_1, x_2, x_3): x_1^2 + x_2^2 + (x_3 -1)^2 \leq \epsilon^2\}$. If we use the two dimensional disks $V = B_{\epsilon}((x_1, x_2, 1)) \cap \mathrm{aff}(U)$ for investigating the interior points, we get $\mathrm{ri}(U) = \{(x_1, x_2, 1) \in \mathbb{R}^3: |x_1| < a, |x_2| < b\}$.
Exercise. Try to imagine $U$ as half of a sphere in $Y = \mathbb{R}^3$. What are $\mathrm{int}(U)$ and $\mathrm{ri}(U)$? If $U$ is a circular arc in $Y = \mathbb{R}^2$, answer the same questions. What does happen if $U$ is a torus in $Y = \mathbb{R}^3$?