Let $k$ be a commutative ring with unity and $E$ a quiver with source and target functions $E^1 \xrightarrow{s,t} E^0$. The Leavitt path algebra of $E$ is the quotient of the path algebra of the double graph $k(E \sqcup E^\ast)$ by the Cuntz-Krieger relations
$$ e^\ast f := \delta_{s,f} \ t(f) \tag{CK 1} $$
for each pair of edges $e,f \in E^1$ and
$$ v = \sum_{e \ : \ s(e) = v}\tag{CK 2}ee^\ast $$
for each regular vertex $v \in E^0$. This means that $s^{-1}(v)$ is non-empty and finite, which means that $v$ is not a sink or an infinite emitter.
This is analogous to the construction of graph $C^*$-algebras, where the original Cuntz-Krieger relations arose, and include several algebras of interest. However, I would like to gain some intuition as to why these relations were considered in the first place.
The original paper of Cuntz seems to relate $(CK1)$ and $(CK2)$ to topological Markov chains and symbolic dynamics, but I've had a hard time following the motivation. I'd really appreciate an explanation for non-experts.
If $A$ is an $n\times n$ matrix of zeros and ones, the Markov space $$ \Sigma _A=\big\{x=(x_k)_{k=1}^\infty \in \{1, 2, \ldots , n\}^\mathbb N: A_{x_k, x_{k+1}}=1, \text{ for all } k\in \mathbb N\big\} $$ may be partitioned by the so called cylinder sets $$ Z_i = \big\{x=(x_k)_k\in \Sigma _A: x_1=i\big\}, $$ defined for each $i$ in $\{1, 2, \ldots , n\}$.
The shift map $$ S: (x_1, x_2, x_3, \ldots ) \mapsto (x_2, x_3, x_4, \ldots ) $$ maps $\Sigma _A$ to itself, and it is evidently injective when restricted to each cylinder $Z_i$. We may then consider the inverse branches of the shift, namely the maps $$ \theta _i : (x_1, x_2, \ldots ) \in S(Z_i) \mapsto (i, x_1, x_2, \ldots ) \in \Sigma _A. $$
Thus, while $S$ deletes the first coordinate of $x$, we see that $\theta _i$ inserts $i$ as a first coordinate of $x$. However insertion cannot always be done since $(i, x_1, x_2, \ldots )$ may sometimes fall outside $\Sigma _A$. In fact notice that the domain of each $\theta _i$ turns out to be $$ S(Z_i) = \bigcup_{{\buildrel {1\leq j\leq n} \over {A_{i, j}=1}}} Z_j. \tag 1 $$ On the other hand the range of $\theta _i$ is precisely $Z_i$.
The original Cuntz-Krieger paper may be considered as a quantization of the Markov shift, where $\theta _1, \theta _2, \ldots , \theta _n$ are replaced by partial isometric operators $S_1, S_2, \ldots , S_n$ on a Hilbert space. The fact that the range projections $$ P_i:= S_iS_i^* $$ are required to add up to one may be seen as an interpretation of the fact that $\Sigma _A$ is partitioned by the ranges of the $\theta _i$. The Cuntz-Krieger relation $$ S_i^*S_i = \sum_{j=1}^n A_{i,j} P_j, $$ if written in the equivalent form $$ S_i^*S_i = \sum_{{\buildrel {1\leq j\leq n} \over {A_{i, j}=1}}} P_j, $$ expresses that the source projections $S_i^*S_i$ behaves somewhat like the domains of the $\theta _i$, as in (1).
It is therefore perhaps unsurprising that one of the main early results about the Cuntz-Krieger algebras is that two Markov shifts are conjugate iff the corresponding Cuntz-Krieger algebras are isomorphic.