The complex structure on a complex manifold pulls back to what's
called a CR structure on any real codimension 1 submanifold.
The structure induced on a submanifold of higher codimension is a CR
structure if a non-degeneracy condition holds.

The CR structure is induced on arbitrary real *CR submanifolds* $M$ in a complex manifold $X$, that is submanifolds for which the intersection $H^{01}$ of the bundle $T^{01}X$ of all $(0,1)$ vectors with the complexified tangent bundle $\mathbb C\otimes TM$ of $M$ is of constant dimension along $M$. In particular, a real codimension $1$ submanifold, i.e. a real hypersurface, is always a CR submanifold.

The induced CR structure of a CR submanifold is then defined by that intersection subbundle $H^{01}\subset \mathbb C\otimes TM$.
(Equivalently, $(1,0)$ vectors can be used instead of $(0,1)$, but the latter is more convenient e.g. in the context of the $\bar\partial$ problem.)

I have never seen the condition being a CR submanifold called a "non-degeneracy condition", and wouldn't do it, because that condition is generally not stable under small perturbation. For instance, a complex line inside $\mathbb C^2$ is a CR submanifold that can be perturbed into a non CR submanifold.

The special property of a codimension $1$ submanifold is that it is always a CR submanifold. The same is not true in general for submanifolds of higher codimension.

It's possible to describe these structures intrinsically, without
reference to an embedding.

Yes, a CR structure on a real manifold $M$ is defined by any complex subbundle $V=H^{01}$ of the complexification $\mathbb C\otimes TM$ satisfying $V\cap \bar V=\{0\}$ and the integrability $[V, V]\subset V$. If only the first condition is assumed, $V$ defines an *almost CR structure*. There is also the intermediate *partial integrability* condition $[V, V]\subset V\oplus \bar V$.

Equivalently, an almost CR structure can be defined without complexification, by a pair $(H,J)$ of a real subbundle $H\subset TM$
and a complex structure $J\colon H\to H$, see e.g. this answer for more details. However, the integrability condition $[V, V]\subset V$ becomes more verbose, when written in terms of $H$ and $J$.

The *CR codimension* of an almost CR structure is defined intrinsically
as the complex codimension of $H^{10}\oplus H^{01}$ in $\mathbb C\otimes TM$, or equivalently, the real codimension of $H$ in $TM$, where
$H = (H^{10}\oplus H^{01}) \cap TM$ and $H^{10}=\overline{H^{01}}$.

- Does CR stand for Cauchy-Riemann, or what?

It stands for both Cauchy-Riemann and Complex-Real.

- What kind of local invariants do CR manifolds have? Are there coordinates around every point that look like a real hyperplane in C^n? Or can there be some curvature or something.

The lowest order invariant, the Levi form, is of the 2nd order. It is possible to choose local coordinates, the only 2nd order terms are the ones from the Levi form.

In contrast to complex structures (corresponding to CR structures of CR codimension $0$), for a general CR structure of positive CR codimension, there are infinitely many higher order local invariants.

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,
for a complete intrinsic normal form, including the non-integrable case.

- Can there be continuous families of CR structures on a given manifold?
If the manifold is compact can these families (mod diffeomorphism) be
infinite-dimensional?

Yes, see above. Also, infinite-dimensional families of non-CR-equivalent
CR structures can be obtained even locally using the Chern-Moser normal form or Cartan connection, see e.g.

Chern, S. S.; Moser, J. K. Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219–271

- I have the impression, just from arxiv postings and seminar titles, of
CR geometry being studied more in analysis than in softer geometric
fields. Is that accurate, and if so what accounts for it?

You can find CR structures in both analysis and "softer" geometry such as compatible CR structures with contact structures, fillable CR structures etc, for instance in this book:

MR3012475 Cieliebak, Kai; Eliashberg, Yakov.
From Stein to Weinstein and back.
Symplectic geometry of affine complex manifolds. American Mathematical Society Colloquium Publications, 59. American Mathematical Society, Providence, RI, 2012.