Standard short exact sequences in Algebraic Geometry

Let $X$ be a scheme of finite type over a field, $Y\subset X$ a closed subscheme and $U=X\setminus Y$ its open complement.
Then we have (Fulton,Prop. 1.8, page 21) for the Chow goups of $k$-cycles the exact sequence $$CH_k(Y)\to CH_k(X)\to CH_k(U)\to 0 $$ This powerful tool immediately implies, for example, that for any closed irreducible hypersurface $Y=Y_d\subset X=\mathbb P^n_k$ of degree $d$ of projective space over a field $k$ we have for $U=\mathbb P^n_k\setminus Y$ : $$\operatorname {Pic}(U)=CH_{n-1} (U)=\mathbb Z/d\mathbb Z$$ allowing us to brag that for every cyclic group we know a smooth algebraic variety whose Picard group is that cyclic group! ( For the infinite cyclic group we show off $\mathbb Z=\operatorname {Pic}(\mathbb P^n_k) $).

The Kummer sequence is another useful one, especially in studying Brauer groups and twisted sheaves: $$ 0 \to \mu_n \longrightarrow \mathbb{G}_m \overset{n}{\longrightarrow} \mathbb{G}_m\to 0 $$ where $\mu_n$ is the set of $n^\text{th}$ roots of unity.

One of the most important sequence is the "tautological sequence". This is very useful when computing Chern classes.

Assume $X$ is the space parametrizing some vector subspace of a fixed vector space $V$. Then, above $x \in X$ there is a corresponding $V_x \subset V$. Then, there will be a corresponding exact sequence $$ 0 \to V_x \to V \to V/V_x \to 0$$

called the "tautological sequence".

The typical example is the Grassmanian $X = G(k,V)$. The sequence is $$ 0 \to S \to V \to Q \to 0 $$

where $V$ is the trivial vector bundle $V \times X$, $S$ is the vector bundle which fiber over $x \in X$ is the corresponding subspace $x \subset V$ and $Q$ is the vector bundle with fiber $Q_x = V/x$.

Similar sequences exist when $X$ is an Hilbert scheme or a flag variety. For more details see the book by Fulton and Harris, "3264 and all that".

Another sequence is pretty useful. If $D \subset X$ is a smooth divisor in a projective variety then there is a long exact sequence $$ \dots \to H^i(X) \to H^i(U) \to H^{i-1}(D) \to H^{i+1}(X) \to \dots $$

which is the algebraic version of the Gysin exact sequence. For a reference this is probably in Fulton's book on intersection theory.