Let $ Fl(m) $ be a complete flag manifold on $\mathbb{C}^m$. We can decompose trivial bundle $Fl(m) \times \mathbb{C}^m $ as ascending sum $ U_1 \subset ... \subset U_m = Fl(m) \times \mathbb{C}^m$ where $U_i = \left\{(u,v) \in Fl(m) \times \mathbb{C}^m: v \in u_i \right\}$ ($u$ is a flag $ u_1 \subset u_2 \subset ...$, $ dim(u_i)= i$ ) We set $x_i = -c_1 ( U_1/U_{i-1})$ ($c_1$ denotes the first Chern class).
Now in Fulton's "Young Tableaux with applications to representation theory" in Proposition 3 on page 161 it is claimed that classes $x_{1}^{i_1} \cdot ... \cdot x_{m}^{i_{m}}$ form a basis of $H^{*}(Fl(m)) $ over $\mathbb{Z}$.
The proof starts with showing that $Fl(m)$ can be constructed as a sequence of projective bundles. We start with $\mathbb{P}(\mathbb{C}^m))$ and its tautological bundle $U_1$. Next we take rank $m-1$ bundle $\mathbb{C}^m / U_1$ over $\mathbb{P}(\mathbb{C}^m))$. Then "we construct $\mathbb{P}( \mathbb{C}^m / U_1 ) \rightarrow \mathbb{P}(\mathbb{C}^m)$. The tautological line bundle on $ \mathbb{P}( \mathbb{C}^m / U_1 ) $ has the form $U_2/U_1 $ for some bundle $U_2$ of rank $2$ with $U_1 \subset U_2 \subset \mathbb{C}^m$ on $\mathbb{P}( \mathbb{C}^m / U_1 )$."
I don't get this part. I'd say $\mathbb{P}( \mathbb{C}^m / U_1 )$ means projective bundle of vector bundle $\mathbb{C}^m / U_1 \rightarrow \mathbb{P}(\mathbb{C}^m)$ but its total space is not a Grassmanian then what does tautological bundle over it mean?
This is a very general construction. Let $E\rightarrow X$ be a vector bundle over $X$. Then there is a bundle over the projectivization $\pi:\mathbb P(E)\rightarrow X$ constructed as follows. For every $x\in X$ a point $[z]\in \mathbb{P}(E_x)=\pi^{-1}(x)$ in the fiber over $x$ defines a line in $\pi^*E_x$, namely the line obtained by complex multiples of $z$, i.e. the line $\mathbb C z$. These lines patch together to form a one dimensional subbundle $\tau$ of the pullback bundle $\pi^* E$. This is the tautological bundle over $\mathbb{P}(E)$.