The relationship between cohomological descent and Lurie's Barr-Beck is exactly the same as the relationship between ordinary descent and ordinary Barr-Back. To put things somewhat blithely, let's say you have some category of geometric objects $\mathsf{C}$ (e.g. varieties) and some contravariant functor $\mathsf{Sh}$ from $\mathsf{C}$ to some category of categories (e.g. to $X$ gets associated its derived quasi-coherent sheaves, or as in SGA its bounded constructible complexes of $\ell$-adic sheaves). Now let's say you have a map $p:Y \rightarrow X$ in $\mathsf{C}$, and you want to know if it's good for descent or not. All you do is apply Barr-Beck to the pullback map $p^\ast:\mathsf{Sh}(X)\rightarrow \mathsf{Sh}(Y)$. For this there are two steps: check the conditions, then interpret the conclusion. The first step is very simple -- you need something like $p^\ast$ conservative, which usually happens when $p$ is suitably surjective, and some more technical condition which I think is usually good if $p$ isn't like infinite-dimensional or something, maybe. For the second step, you need to relate the endofunctor $p^\ast p_\ast$ (here $p_*$ is right adjoint to $p^\ast$... you should assume this exists) to something more geometric; this is possible whenever you have a base-change result for the fiber square gotten from the two maps $p:Y \rightarrow X$ and $p:Y \rightarrow X$ (which are the same map). For instance in the $\ell$-adic setting you're OK if $p$ is either proper or smooth (or flat, actually, I think). Anyway, when you have this base-change result (maybe for p as well as for its iterated fiber products), you can (presumably) successfully identify the algebras over the monad $p^\ast p_\ast$ (should I say co- everywhere?) with the limit of $\mathsf{Sh}$ over the usual simplicial object associated to $p$, and so Barr-Beck tells you that $\mathsf{Sh}(Y)$ identifies with this too, and that's descent. The big difference between this homotopical version and the classical one is that you need the whole simplicial object and not just its first few terms, to have the space to patch your higher gluing hopotopies together.
Suppose we are given some category (or higher category) of "spaces" in which
each space $X$ is equipped with a fiber, i.e. a category $C_X$ of objects
of some type over it.
For example, a space can be a smooth manifold and the fiber
is the category of vector bundles over it;
or a space is an object of the category dual to the category of rings
and the fiber is its category of left modules.
Given a map $f: Y\to X$, one often has an induced
functor $f^* : C_X\to C_Y$ (pullback, inverse image functor,
extension of scalars).
The basic questions of classical descent theory
are:
When an object $G$ in $C_Y$ is in the image via $f^*$ of some object in $C_X$ ?
Classify all forms of object
$G\in C_Y$, that is find all $E\in C_X$ for which $f^*(E)\cong G$.
Grothendieck introduced pseudofunctors and
fibred categories
to formalize an ingenious method to deal with descent questions.
He introduces additional data on an object $G$ in $C_Y$
to have a chance of determining an isomorphism class of an object in
$C_X$. Such an enriched object over $X$ is called a ``descent datum''.
$f$ is an effective descent morphism if the morphism $f$
induces a canonical equivalence of
the category of the descent data (for $f$ over $X$) with $C_X$.
It is a nontrivial result that in the case of rings and modules,
the effective descent morphisms are preciselly
pure morphisms of rings. Grothendieck's
flat descent theory tells a weaker result
that faithfully flat morphisms are of effective descent.
In algebraic situations one often introduces a (co)monad $T_f : C_X\to C_X$
(say with the multiplication $\mu: T_f \circ T_f \to T_f$)
induced by the morphism $f$.
The category of descent data is then nothing else than the
Eilenberg-Moore category $T_f-\mathrm{Mod}$ of (co)modules (also called (co)algebras) over $T_f$.
Then, by the definition, $f$ is of an effective descent if and only if
the comparison map (defined in the (co)monad theory)
between $C_X$ and $T_f-\mathrm{Mod}$ is an equivalence.
Several variants of Barr-Beck theorem give conditions
which are equivalent or (in some variants) sufficient to
the comparison map for a monad induced by a pair of adjoint functors
being an equivalence. Generically
such theorems are called monadicity (or tripleability) theorems.
One can describe most of (but not all)
situations of 1-categorical descent theory via the monadic approach.
There are numerous generalizations of monadicity theorems, higher cocycles and descent, both in monadic and in fibered category setup
in higher categorical context (Giraud, Breen, Street, K. Brown, Hermida, Marmolejo, Mauri-Tierney, Jardine, Joyal, Simpson, Rosenberg-Kontsevich, Lurie...); the theory of stacks, gerbes and of general cohomology is almost the same as the general descent theory, in a point of view.
For examples, it is better to consult the literature. It takes a while to treat them.
Best Answer
For question 1, see the comment above.
Collecting the answers to question 2:
"Community wiki" post, feel free to modify.