You are right. These (2,0) theories are theories that do not admit a Lagrangian description in the UV (generally theories not admitting a Lagrangian description must be conformal field theories) and can be ADE classified. These theories are the highest dimension non-trivial CFTs we know. You can study such theories by making various types of compactifications string dualities and so on.

The basic idea begins by considering strings whose tension varies. This variation should take place in a space $Z$ with two extra dimensions and then embed this space into a four-dimensional (more familiar) ambient space $X$. Let $\Sigma$ be a two-sheet cover of $Z$ (remember how $\Sigma$ is the Seiberg-Witten curve). Now, imagine that $Z$ is covered by these two sheets of $\Sigma$, separated in the additional directions of $X$. Furthermore let us consider a membrane that is extended along the 2 extra dimensions of $Z$ and 1time-direction (of $X$). We can denote these coordinates as $(z,x)$ obviously with $z\in Z, \, z \in X$. Of course, $x$ is a complex coordinate and the two sheets define $x_1(z)$ and $x_2(z)$. We can consider what the tension of the membrane that is hanging between the two sheets is. It turns out that the constant tension is given by $|dx|\wedge |d\log z|$ and this membrane can be thought of as a string whose tension is a function of $z$ and satisfies $$T_{\text{string}}(z)\geq \left| \lambda_1 - \lambda_2 \right|,$$ where $\lambda_i = \left| x_i \frac{dz}{z} \right|$. Now, it turns out (to be fair I do not know a lot about it) that in M-theory appear various kinf of higher-dimensional objects with similar properties. The typical example is to consider M-theory and of course M5-branes and M-2 branes (the only type of branes of M-theory) in the product space4 $\mathbb{R}^{3,1} \times X \times \mathbb{R}^3$. Put a brane on the subspace $\mathbb{R}^{3,1} \times \Sigma \times \{0\}$. Here, $\Sigma \subset X$ while 0 is the origin of $\mathbb{R}^3$. The resulting theory is a 4d theory. Now put one M2-brane with endpoints on the M5-brane on the space $\mathbb{R}^{0,1} \times D \times \{0\}$. Here $\mathbb{R}^{0,1} \subset \mathbb{R}^{3,1}$ (i.e. the worldline of a particle in 4d space) while $D\subset X$ with boundary on $\Sigma$. If we consider the above system as a 6d one then we do get the 6d $\mathcal{N}=(2,0)$ theory, e.g. on $\mathbb{R}^{3,1} \times Z$. By going further, we find out that this theory has a spectrum with both electrically and magnetically charged objects originating from the M-brane. It turns out that htis theory has a self-duality of the form $G = \star_6 G$ where $(\star G)_{\alpha \beta \gamma}= \epsilon_{\alpha \beta \gamma \mu \nu \sigma} G^{\mu \nu \sigma}$ (analog of the electric field in 6d). By placing the theory on a torus with coordinates $x_5,x_6$ you can consider strings wrapping these directions. Then, the field strength of the electric and magnetic fields are given by $F_{E \, \mu \nu} = G_{6\mu \nu}$ and $F_{D \, \mu \nu } = G_{5\mu \nu}$. The duality arises from the previous duality between the $G$'s. Namely in 4d it is $F_{D} = \star_4 F_E$. Reminds you of something, right??

Ok, I 've written this in my break of this horrible and annoying calculation I am trying to do so sorry for not being more thorough. I am sure that Urs will have much more to say. The best place ot start ooking for references is the nLab article and the references therein. This reference might be interesting too. Also, I think Vafa (and maybe some other people) had nice talks on String 14 on the subject. Also try to look info for the very related AGT conjecture.