Conceptual Differences among Galerkin Methods I have a conceptual question about numerical methods for second-order elliptic partial differential equations. 
What is the difference among finite element, continuous finite element, discontinuous finite element, continuous galerkin and discontinuous galerkin methods? Are some of them just the same thing?
 A: All numerical methods for approximating the solution of differential equations address the infinite dimensionality of function spaces in which an exact solution is postulated by constructing a finite dimensional analog which can provide a suitable approximation.  This is true of finite difference methods as well as finite element methods.
Galerkin methods are useful in approximating solutions to systems of differential equations (and sometimes more general problems) that have weak or variational formulations.
A typical example is a second-order elliptic PDE in a simply connected domain $\Omega$ of the plane, with homogeneous Dirichlet boundary conditions:
$$ a u_{xx} + b u_{xy} + c u_{yy} = f \;\; \text{ on } \Omega $$
$$ u = 0 \;\; \text{ on } \partial \Omega $$
which has a weak formulation obtained by multiplying by a "test function" $v$ and performing integration by parts:
$$ A(u,v) = F(v) \;\; \forall v $$
where $A(\cdot,\cdot)$ is a bilinear form (on a suitable function space) and $F$ is a linear functional on that space.
While the "classical" or strong solution to the PDE may be sought in a function space such as $\mathcal{C}^2(\Omega)$, requiring two continuous derivatives (up to the boundary), the weak formulation allows us to "split" the order of differentiation between the "trial function" $u$ and the "test function" $v$.  Reducing the required "smoothness" of derivatives (here, from second-order continuous derivatives to square integrable first-order derivatives) allows us to work with fairly simple classes of functions in approximating the solutions to the classical problem.
That is, if $V = \mathcal{W}_0^1(\Omega)$ is the Sobolev space of functions on $\Omega$ with square integrable first-order derivatives and "trace" zero on the boundary, then a finite dimensional subspace $V_n \subset V$ may be used to get a discrete Galerkin approximation scheme by solving the weak formulation of the problem on $V_n$:
$$ A(\hat{u},\hat{v}) = F(\hat{v}) \;\; \forall \hat{v} \in V_n $$
This leads to a system of linear equations to solve, given a basis for $V_n$, of size $n\times n$ where $\dim(V_n) = n$.  Ellipticity of the PDE implies that the bilinear form $A(\cdot,\cdot)$ is coercive, and this in turn implies that the discretized solution (finitely many unknowns) is a quasi-optimal approximation of the weak solution $u \in V$.  Here quasi-optimal means that the error of approximation $||u - \hat{u}||_{\mathcal{W}^1}$ is within a constant factor of the error of the best approximation available in $V_n$.  [Note: The constant factor will depend on the coefficients of the elliptic PDE and on the domain $\Omega$.]
Finite element methods are a subset of these in which the approximating subspaces $V_n$ are constructed by geometric subdivision of $\Omega$ and trial/test functions that are defined piecewise on those subdivisions ("elements") in a way that achieves the required smoothness of functions in $V$.  For example we might triangulate the domain $\Omega$ and work with approximating functions that are piecewise linear (but continuous) with respect to that triangulation.
Now there are many variations on this theme.  What we've described above are conforming Galerkin (resp. finite element) methods, in which the subset relation $V_n \subset V$ of the discrete and infinite dimensional function spaces holds.  Some of the terms asked about in the Question concern non-conforming approximation schemes, where we may allow smoothness of the functions to be further reduced. 
Thus Discontinuous Galerkin methods are in the category of non-conforming approximation schemes (discretizations).  In practice for elliptical problems these are also implemented as hybrid methods, meaning that the trial functions are two-part combinations of functions on "elements" (such as triangles) with functions on "facets" (such as edges of triangles).  In this case the integration by parts cannot be carried out over the entire domain $\Omega$, and a means must be found to work with a variational formulation based on partial integration by parts on the individual elements.
It is impossible to even scratch the surface of choices available for trial functions and test functions across the various applications of numerical PDEs.  However in many cases the use of piecewise polynomials is preferred because of the simplicity of performing integration with them. 
I will finish with a broad and poorly justified generalization.  These topics are intriguing because the analysis of discrete approximations to continuous functions is informed by geometric constructions (mesh generation) whose tradition goes back to Euclid and Archimedes.
