Typically to describe a ring or an algebra to someone, you would present them with a full set of generators and provide relations between them under the appropriate operations. Things are not as simple or as courteous with cluster algebras...
To describe a cluster algebra, you first provide an initial seed that contains some generators. You then provide rules for how you can mutate a seed to obtain a new seeds. You repeat this (possibly infinite) process starting with the initial seed. You then collect all the elements from all seeds into a set and generate an algebra from them.
These rules for mutation can be encoded in a quiver associated to the initial seed. This quiver is also mutated between each seed.
Given $k,n \in \mathbb{N}$, with $k \leq n$, the Grassmannian $Gr(k,n)$ is defined as the set of k-dimensional subspaces in $\mathbb{C}^n$, i.e. $$Gr(k,n) = \{V \leq \mathbb{C}^n \mid \mathrm{dim}\,V = k\}$$ How can we describe an element of a Grassmannian? Let's consider $Gr(2,4)$. We can take a max rank $2 \times 4$ matrix $$A = \begin{pmatrix} a & b & c & d \\ e & f & g & h \end{pmatrix}$$ and consider its row span. Since A has rank 2, its row span is a 2-dimensional subspace of $\mathbb{C}^4$, i.e. it determines an element $V \in Gr(2,4)$.