See e.g. :
[page 7] Definition 2.1.2 A sequent is an expression ($Γ \vdash \psi$) where $\psi$ is a statement (the conclusion of the sequent) and $Γ$ is a set of statements (the assumptions of the sequent). We read the sequent as ‘$Γ$ entails $\psi$’. The sequent ($Γ \vdash \psi$) means
There is a proof whose conclusion is $\psi$ and whose undischarged assumptions are all in the set $Γ$.
After having introduced the usual Natural Deduction rules for the propositional conncetives, we have :
[page 54] Definition 3.4.1 Let $σ$ be a signature. Then a $σ$-derivation or, for short, a derivation is a left-and-right-labelled tree (drawn branching upwards) such that the following hold:
(a) Every node has arity $0, 1, 2 or 3$.
(b) Every left label is either a formula of $LP(σ)$, or a formula of $LP(σ)$ with a dandah [i.e. a "crossed formula"].
(c) Every node of arity $0$ [i.e. a leaf] carries the right-hand label (A).
(d) If $\nu$ is a node of arity $1$, then one of the following holds:
details regarding the rules follow;
(e) If $ν$ is a node of arity $2$, then one of the following holds:
(f) If $ν$ is a node of arity $3$, then
(g) If a node $\mu$ has left label $\chi$ with a dandah, then $\mu$ is a leaf, and [...]
The conclusion of the derivation is the left label on its root, and its undischarged assumptions are all the formulas that appear without dandahs as left labels on leaves. The derivation is a derivation of its conclusion.
Another definition is into :