In addition to other excellent answers, I would like to approach from another angle.

## Almost CR structures

A most general *almost CR structure* (i.e. of arbitrary codimension) on a real manifold $M$ is given by a complex subbundle

$$V\subset \mathbb C\otimes TM,$$

of the complexified tangent bundle of $M$, satisfying $V\cap \bar V=\{0\}$, where the conjugation on $V$ is induced by the standard conjugation on $\mathbb C\otimes TM$.

This definition is equivalent to its "real version" given by the pair $(H, J)$, where $H\subset TM$ is a real subbundle, and $J\colon H\to H$ is a complex structure on $H$. Indeed, given $(H, J)$ as above, the corresponding $V$ is the so-called $(0,1)$-subbundle given by

$$ V= H^{01} := \{ L\in \mathbb C\otimes H: JL = -iL \} = \{ X + iJX: X\in H\}.$$

Vice versa, every $V$ as above yields

$$H:= (V +\bar V)\cap TM,$$

and $J\colon H\to H$ is uniquely determined by the identity

$$X + iJX\in V, \quad X\in H.$$

(An equivalent description can be given via $(1,0)$ instead of $(0,1)$ vectors, but the former is preferred e.g. in the context of the $\bar\partial$ equation.)

## CR codimension

An important invariant of an almost CR structure is its *CR codimension*,
given by the complex codimension of $V\oplus \bar V$ in $\mathbb C\otimes TM$ or, equivalently, by the real codimension of $H$ in $TM$.
The almost CR structures of codimension $0$ are precisely the almost complex structures, and those of codimension $1$ are the ones induced on real hypersurfaces in almost complex manifolds. Important examples of almost CR structures of higher codimension are given by real Lie group orbits, for instance, boundary components of bounded symmetric domains may have arbitrarily high CR codimension.

## Almost CR structures as G-structures

To regard an almost CR structure $V$ as a G-structure,
consider the subbundle of all frames in $TM$, consisting of all CR isomoprhisms from the *flat model* $\mathbb C^n\oplus \mathbb R^m$ onto $TM$. Then the group $P$ of all linear CR isomoprhisms of the flat model acts transitively on these frames at each point, and thus defines an $P$-structure on $M$. Vice versa, any such $P$-structure corresponds to an unique almost CR structure.

This construction works, in particular, for almost complex structures, since it is the special case of an almost CR structure corresponding to CR codimension 0.

## Induced almost CR structures on real submanifolds of almost complex manifolds

For any real submanifold $M$ in an almost complex manifold $C$,
there is canonical almost complex structure on each complex tangent space
$H_p:= T_pM\cap J T_pM$ at a point $p\in M$, where $J$ is the complex structure of $C$.
If $H_p$ is of constant dimension, $M$ is called CR submanifold of $C$
with the induced almost CR structure. If $M$ is a real hypersurface, it is always a CR submanifold.

## Almost CR G-structure as a reduction of an almost complex G-structure

One way to see such reduction, is to consider a CR submanifold $M\subset C$ of an almost complex manifold $C$, such that $M$ *is generic in* $C$,
which means $TM+JTM = TC|_M$, i.e. $TM$ needs to span the full tangent space to $C$ over $\mathbb C$. That genericity condition guarantees that any CR frame can be uniquely extended to a complex frame in $C$.
Then the bundle of all CR frames on $M$ (defining the corresponding $G$-structure) is canonically identified with a subbundle of all complex frames on $C$ resticted to $M$, and hence can be seen as a reduction of the corresponding almost complex G-structure.

## Integrability

An almost CR structure $V\subset \mathbb C\otimes TM$ is *(formally) integrable* if the subbundle $V$ is closed under Lie brackets. In contrast to almost complex structures, such formal integrability does not guarantee the existence of sufficiently many local CR functions (implying a local CR embedding into a complex manifold), which is a stronger property sometimes called integrability without the "formal" adjective. Most commonly CR structures are defined as formally integrable almost CR structures, whereas the latter ones are called locally embeddable.

Note that integrability played no role in the above discussion and comes as additional property. I suppose that was the confusion mentioned in the question.

Also note that, in contrast to complex structures, integrability of almost CR structures does not imply any ``flatness'' such as CR equivalences with their flat models.

## Nondegeneracy conditions

The most well-known nondegeneracy condition is the one for the Levi form,
which can be defined for any almost CR structure (or any codimension).

See e.g. my paper, Normal forms for almost non-integrable CR structures, Amer. J. of Math., 134 (2012), no. 4, 915-947, also available on the arxiv.org.

In particular, if $M$ is of CR codimension $1$ (i.e. of hypersurface type), adding a nondegenerate Levi tensor allows to reduce the almost CR G-structure to the one admitting a canonical Cartan connection, as more extensively described by Andreas Cap in his answer.

There are also many natural generalizations of the Levi-nondegeneracy condition in various directions, where many details can be found in this book:

MR1668103 (2000b:32066)
Baouendi, M. Salah(1-UCSD); Ebenfelt, Peter(S-RIT); Rothschild, Linda Preiss(1-UCSD)
Real submanifolds in complex space and their mappings.

Princeton Mathematical Series, 47. Princeton University Press, Princeton, NJ, 1999. xii+404 pp. ISBN: 0-691-00498-6

## Contact and almost contact structures

In contrast to almost CR structures, contact structures already contain a nondegeneracy condition in their definition. The corresponding almost contact structure is then defined by adding a nondegenerate symplectic form to the hyperplane field, unrelated to the one given by the Lie bracket. That can be equivalently described by appropriate G-structure
in a similar way as for almost CR structures described above.