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Definition & Example
 A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(2n,0)$ consisting of $n$ up steps of the form $(1,1)$ and $n$ down steps of the form $(1,1)$ which never goes below the xaxis $y=0$. Denote all Dyck paths of size $n$ by $\mathfrak{D}_n$.
Equivalently, a Dyck path of semilength $n$ can be seen as

a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ up steps of the form $(1,0)$ and $n$ down steps of the form $(0,1)$ which never goes below the diagonal $y=x$.

a Dyck word, this is a word in $\{0,1\}$ such that any prefix contains at least as many $1$'s as it contains $0$'s. Seeing a $1$ as an opening bracket and a $0$ as a closing bracket, Dyck words can be seen as wellformed bracketing systems.
the 5 Dyck paths of size 3  
[1,0,1,0,1,0]  [1,0,1,1,0,0]  [1,1,0,0,1,0]  [1,1,0,1,0,0]  [1,1,1,0,0,0] 
 There are $\operatorname{Cat}(n) = \frac{1}{n+1}\binom{2n}{n}$ Dyck paths of semilength $n$, see OEIS:A000108. This can for example be seen using the reflection principle.
Properties

A Dyck path $D$ of size $n+1$ can be decomposed into a Dyck path $D_1$ of size $k$ and a Dyck path $D_2$ of size $nk$, where

$(2k+2,0)$ is the first touch point of $D$ at the xaxis,

$D_1$ is the prefix of $D$ from $(1,1)$ to $(2k1,1)$ never going below the line $y=1$, and where

$D_2$ is the suffix of $D$ from $(2k,0)$ to $(2n,0)$.
This yields the recurrence
Remarks

There is a natural extension of Dyck paths to $m$Dyck paths which can be seen as lattice paths from $(0,0)$ to $(mn,n)$ that never go below the diagonal $y = \frac{1}{m} x$. The number of $m$Dyck paths is given by the FussCatalan number $C^{(m)}(n) = \frac{1}{mn+1} \binom{(m+1)n}{n}$ [Kra89].

Dyck paths and $m$Dyck paths can be seen as type A instances of a more general combinatorial object for root systems, namely positive chambers in the (generalized) Shi arrangement [Ath04].

Dyck paths of semilength $n$ are in canonical bijection with Nakayama algebras with linear quiver and $n+1$ simple modules.
References
 [Ath04] C.A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc., 36 (2004), pp. 294392
 [Kra89] C. Krattenthaler, Counting lattice paths with a linear boundary II, Sitz.ber. d. ÖAW Math.naturwiss. Klasse 198 (1989), 171199
Sage examples
Technical information for database usage
 A Dyck path is uniquely represented as a Dyck word.
 Dyck paths are graded by the semilength.
 The database contains all Dyck paths of semilength at most 8.