Everything that can be expressed in terms of the first fundamental form (in a coordinate-independent way) gives an intrinsic quantity. An amazing property of the Gaussian curvature is that being defined as the determinant of the shape operator (i.e. using the "outside" geometry) it turns out to be expressible only as a combination of the components of the first fundamental form and their partial derivatives (see the Brioschi formula).
There are many things that you can make using the first fundamental form, for instance, we can measure lengths of smooth curves. These ares obviously intrinsic quantities.
Polynomial expressions involving components of the first fundamental form and all its partial derivatives of finite order are called natural if the form of the expression does not depend on the choice of (normal) coordinates. Examples are the Levi-Civita connection and the Riemann tensor (and of course the first fundamental form itself, which is also known as the intrinsic metric). It turns out that all possible natural tensors are obtained as iterated covariant derivatives of the Riemann curvature tensor and their contractions. Details see in the paper of D.B.A. Epstein "Natural tensors in Riemannian geometry", e.g. here.
On the other hand, the second fundamental form (which is an appearance of the shape operator) cannot be measured without an immersion: you need a unit normal field along the surface.
The terms "intrinsic" and "extrinsic" are confusing when trying to be defined in introductions to differential geometry, and for good reason: the standard definition of surfaces at this level float between things like "a subset of $\mathbb{R}^3$ such that etc etc".
A surface is a topological space. A topological space with a particular set of properties, and its definition generalizes to manifolds, which are "surfaces" of higher dimensions. A Riemannian manifold is a manifold together with a Riemannian metric, which is a kind of object defined on a space associated to the manifold. It is common to denote a Riemannian manifold by $(M,g)$, where $M$ is the manifold (and its underlying topology, which we usually omit from notation) and $g$ is the metric. I did not expose precisely what means to be a Riemannian manifold, but nowhere in the definition I'm alluding to is supposed that a Riemannian manifold is inside some $\mathbb{R}^n$.
Before proceeding, an analogy may go well (although, as all analogies, it is not perfect). Consider your nickname: "bubba", as defined by a concatenation of characters. Do you need a paper in order to conceive your name? Or a blackboard? Your nickname has an abstract existence on itself. If I were to ask, say: "How big is the 'u' on your name?", this question would make little sense. It depends on how you write it on paper. The length of the letters is an extrinsic property. However, having five letters is an intrinsic property: it doesn't matter how/where it is written, it is a result of how your name is defined.
Now, moving on. We then usually say that a property of a Riemannian manifold $(M,g)$ is intrinsic if it is a byproduct only of the topology on $M$, and $g$. One example of an intrisic property is the fact that any smooth function on the torus $T^2$ has at least two critical points (in fact, the lower bound is a little bigger). This is a consequence of the fact that $T^2$ is compact. You may say that we know that $T^2$ is compact since it is a subset of $\mathbb{R}^{3}$, but what if I told you that my $T^2$ is $S^1 \times S^1$? This lives inside $\mathbb{R}^4$ instead, and is completely different setwise than what you imagine as a standard "doughnut torus". If I said that my $T^2$ is the square with convenient identifications, then this $T^2$ isn't in any $\mathbb{R}^n$ setwise-ly speaking. Intrinsic properties receive this name because they do not depend on how you envision them, only on the structure the spaces have.
This has a lot of theoretical and practical applications. But I think there is a reason why this terminology is not so abundant in all mathematics, and it is due to the practical applications of geometry. For example, Gauss's result that the curvature is an intrinsic information is marvelous: it says that something that you can define using the way that a normal vector field varies (and a normal vector field clearly depends on how you put your surface in space) can be computed directly through measures which are related to the tangent space and the Riemannian metric (namely, the first fundamental form - it may not be clear how the tangent space is something intrinsic if you think about it geometrically, so I suggest you look up for one of the abstract definitions of tangent space), and therefore are intrinsic - it doesn't matter how you are "inside" space. In fact, it doesn't matter that you are "inside" space.
For instance, this has a lot of importance in general relativity (although the setup is not exactly Riemannian manifolds): you may have heard that spacetime is curved. This terminology can be quite confusing, and sometimes people try to explain the concept by analogy with how balls curve a rubber sheet etc. However, a big part of the success of the theory is precisely that we don't need that our space is curved inside anything: we don't need to ask "what is outside", and it doesn't make sense a priori (and it should not). It is "curved" in a way that we can define only by means of itself, making it measurable and not a pseudo-science concept.
Now, back to the beginning, it is perfectly understandable that the "intrinsic/extrinsic" duality (and its usefulness) is a little cloudy if you do not know the abstract definitions. If the above discussion does not clear some things up, I think it may be wise to wait for (or go for) the abstract definitions.
Best Answer
The mean curvature of a surface at a point is an extrinsic quantity. The Gaussian curvature is an intrinsic quantity. The mean curvature is the average of the two principal curvatures; the Gaussian curvature is the product of the principal curvatures. The principle curvatures are extrinsic quantities.
It is not at all obvious that the Gaussian curvature is an intrinsic property, but it is.
You asked (in a comment) for a formal definition of extrinsic vs. intrinsic. Here goes, at least for scalar-valued functions on surfaces. Suppose we have a definition for such a function, so for any surface $S$ the definition gives us a function $f_S$ on $S$. For an intrinsic function (like Gaussian curvature), the following holds:
For example, let $S$ be a (flat) rectangle, and let $T$ be half a cylinder, obtained by bending the rectangle along one axis, so that that axis becomes a semi-circle of radius $r$. So we have a bijection $\psi:S\rightarrow T$ (the "bending map"). Let $p$ be a point in the middle of the rectangle. The principal curvatures at $p$ are 0 and 0, since any two lines through $p$ are straight. The principal curvatures at $\psi(p)$ are 0 and $1/r$. The Gaussian curvature is 0 in both cases, but the mean curvature is 0 on the rectangle and $1/2r$ on the semi-cylinder. (Of course, this example doesn't prove in general that the Gaussian curvature is intrinsic, but it does show that mean curvature is not intrinsic---i.e., extrinsic.)
Caveat: the definition above is clumsy and crude (though not wrong) in ways that would take too long to explain fully. Briefly, intrinsic vs. extrinsic still makes sense locally. (Curvature after all can be defined locally.) Also, dealing only with scalar-valued functions is too restrictive. However, we need a coordinate-independent definition of tensors for a "good" definition, which is another whole story.
The definition for general manifolds is pretty much the same: isometric invariants. In other words, just replace the word "surface" with "$n$-dimensional manifold".
You also asked for a reference. I looked in a few books, but they don't provide formal definitions of intrinsic vs. extrinsic. Here is a typical discussion from Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes:
You'll also find good discussions in Gravitation (Misner, Thorne, and Wheeler, $\S21.5$, "Intrinsic and Extrinsic Curvature"), and a historical treatment in Ch.4 of Wells, Differential and Complex Geometry: Origins, Abstractions and Embeddings.
The basic idea is that anything defined using only the metric (and the differential manifold structure) must be an isometric invariant. That's why you'll find the phrase "intrinsically defined" often used.
Finally, let me address one possible source of your confusion. I've been talking about "intrinsic" vs. "extrinsic" in the context of differential geometry, and for local properties (like curvature). But the terms are generally used informally, to contrast properties that depend only on the "abstract manifold" vs. an imbedding of the manifold. The other answer to your question (by gandalf61) gives a couple of good topological illustrations. The knottedness property depends on the imbedding of the circle in $\mathbb{R}^3$. Orientability on the other hand is a homeomorphism invariant, depending only on the topology of the space.