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I read in section 2 of this paper :

"There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram."

What exactly is here referring to here? What is this process called to descent from a group to smaller groups and how does it work?

Tim
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If you delete a node from a Dynkin diagram associated to a group $G$, you get a group which is a subgroup of $G$. In fact, it is a parabolic subgroup. (I can't come up with a reference for this, but I'm pretty sure that standard texts will state this fact. For example see Carter Simple groups of Lie type or Wilson The finite simple groups.) I don't know if this process has a name, it is a well known fact in the theory of (finite and algebraic) groups of Lie type, though.

For example you have $$A_1<A_2<...<A_7<E_8$$

As further information, check the wikipedia page on root datum

A connected split reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.

rafforaffo
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If you delete an outer vertex of a Dynkin diagram and its connecting edges, then you always obtain a Dynkin Diagram of rank one lower, thus representing again a simple group.
The above Wikipedia page already gives an answer: "Some inclusions of root systems can be expressed as one diagram being an induced subgraph of another, meaning "a subset of the nodes, with all edges between them". This is because eliminating a node from a Dynkin diagram corresponds to removing a simple root from a root system, which yields a root system of rank one lower."

Dietrich Burde
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    Thanks for your answer! One thing that is confusing me is that if we remove a node from the, say $SU(5)$, Dynkin diagram is the result $SU(3)\times SU(2) $ or $SU(3)\times SU(2) \times U(1)$, because so far I have read both and I thought maybe there is some systematic approach that clarifies such issues. The Dynkin diagrams are on the level of Lie algebras, and maybe there is a reason why we always get an extra $U(1)$ if we remove a node on the level of groups? – Tim Apr 17 '15 at 08:54