My understanding is that they're named "Euler systems" because that "Frobenius acting on T" in the definition (line 4, p. 22 of Rubin's book) is an "Euler factor" as in Euler's product decomposition of the Riemann zeta function.

The two easiest examples of Euler systems are the so-called cyclotomic units (not the roots of unity, but slightly more complicated, but still classical, beasts built out of them) and the elliptic units (built out of torsion points on CM elliptic curves). Not coincidentally, these are related to the only two types of global fields that we know how to explicitly construct abelian extensions of. Then there is Kato's more complicated Euler system of Heegner points [EDIT - This phrase was wrong- please see post below!], and Kolyvagin's Euler system of Stickelberger elements. All these are described in Rubin's book, but if you haven't seen them before, it might help to have more references.

If group cohomology is still new to you, the cyclotomic units are the best Euler systems to start with, since you don't need Galois cohomology to define them. (Norm-coherent units map to corestriction-coherent cohomology classes under the Kummer map, which is why these global units form an Euler system in the sense of Rubin's book). Rubin's appendix in Lang's republished books Cyclotomic Fields I and II is easier reading for this. Rubin's Inventiones paper on the main conjecture for CM elliptic curves also contains a nice introduction to the technique.

The application of Euler systems to number theory is the following: Euler systems allow us to bound Selmer groups of p-adic Galois representations. These generalize the Selmer group attached to an abelian variety, the ideal class group, and other objects of arithmetic interest. Bounding them is good because it allows us to prove Iwasawa Main Conjectures, which link the behavior of Selmer groups to p-adic L-functions and encompass basically every classical arithmetic tool for computing p-parts of special values of L-functions.

Washington's book on cyclotomic fields explains this in the cyclotomic example, and the spirit and flavor of the general theory is the same. Coates and Sujatha's book Cyclotomic Fields and Zeta Values is also an excellent introduction to the Euler system technique in the cyclotomic setting.