Origin of the flag variety When and where were flag varieties (more generally $G/P$'s) first introduced? What was the original motivation to study it?
To my knowledge, I'd say an early motivation to study the Grassmannian was to formalize Schubert calculus, and their use in the theory of characteristic classes later. I also know flag varieties are essential for the splitting principle but surely they did appear before.
 A: Question: "When and where were flag varieties (more generally G/P's) first introduced? What was the original motivation to study it?"
Answer: Maybe a Historian should give a precise answer to this, but words such as "Grassmannian" and "Plucker" are the names of mathematicians involved in the study of these varieties in the 1900s.
It is a fact that for any semi simple algebraic group $G$ over a field $k$ of characteristic zero and any parabolic subgroup $P \subseteq G$ the quotient $\pi:G \rightarrow G/P$ is a smooth projective variety - the flag variety of $P$.
Given any invertible sheaf $L \in Pic^G(G/P)$ it follows its global sections $H^0(G/P, L)$ (if non-empty) is a finite dimensional irreducible $G$-module and all finite dimensional irreducible $G$-modules can be constructed in this way (the Borel-Weil-Bott theorem). There is also the study of regular holonomic $D_{G/P}$-modules $E$ on $G/P$ (the Bernstein-Beilinson notion $D$-affinity) and relation to generalized Verma modules. This is a much studied topic: The relation between the representation theory of $G$ and the geometry of $G/P$. The Bezout theorem is a theorem about the intersection of curves in $\mathbb{P}^2_k$, and this goes back to the 1700s and 1800s and Newton and Bezout. This theorem does not hold for plane affine curves.
Comment: "Thank you. I think D-modules on G/P comes much after the introduction of the flag varieties. I was interested in the original motivations to study flag varieties, so I don't think this is an answer to my question. – student"
Answer: Yes, D-modules were introduced in the 1960s/70s. One of the first occurrences of the projective  plane was with the "Bezout theorem", and as mentioned this theorem is not true for the affine plane. The theorem is non-trivial since it speaks of multiplicities: If $C_1,C_2 \subseteq \mathbb{P}^2_k$ are two plane projective curves of degrees $d_1,d_2$ where
$$C_1 \cap C_2=\{P_1,..,P_l\}$$
are their common points (with intersection multiplicity $m_i$) it follows
$$ \sum_i m_i =d_1d_2.$$
Hence the theorem speaks of multiplicity. You may describe $\mathbb{P}^2_k$ as $SL(3,k)/P$ for a parabolic subgroup $P$, hence the projective plane is an example of a "flag variety $G/P$": It is a parameter space parametrizing lines $l$ in $k^3$.
This is what the flag variety does in general: If $1 \leq d_1<d_2<\cdots d_k< d$ is a set $\underline{D}$ of integers, there is the flag variety $\mathbb{F}:=\mathbb{F}(\underline{D},k^d)$. This is a smooth projective variety parametrizing flags $E_i$ in $k^d$ of type $\underline{D}$. There is a parabolic subgroup $P \subseteq SL(d,k)$ and an isomorphism $\mathbb{F}(\underline{D},k^d)\cong SL(d,k)/P$. There is a 1-1 correspondence between
the set of $k$-rational points $\mathbb{F}(k)$ of $\mathbb{F}$ and the set of flags in $k^d$.
