An early result in classical Iwasawa theory is Kummer's criterion, which relates special values of the Riemann zeta-function to unique factorization in certain rings obtained by adjoining roots of unity to Z. More specifically, fix a prime p and let w_p be a primitive pth root of unity. Euler proved several centuries ago that the zeta-function takes on rational values at negative integers, and Kummer's criterion states that the class number of Q(w_p) is divisible by p if and only if there is some nonnegative integer m such that p appears in the numerator of the rational number Zeta(-m). The smallest such prime is 37: we have Zeta(-31)=37*683*305065927/(2^6*3*5*17), hence the class number of Q(w_37) is nontrivial (so Z[w_37] does not have unique factorization).
Kummer's criterion provides a bridge between zeta-values and algebraic invariants of cyclotomic fields. Modern Iwasawa theory proposes a much deeper connection between p-adic L-functions (analytic functions on the p-adic unit disc interpolating classical L-values) and certain Galois modules -- class groups of number fields (related to unique factorization) and Selmer groups of elliptic curves (related to the structure of rational points) being primary examples. The analyticity of p-adic L-functions allow them to be viewed as power series with coefficients in some p-adic field. Attached to these power series are Iwasawa invariants mu and lambda -- they are integers which measure, respectively, the divisibility of the power series by p and its number of roots as a function on the open p-adic unit disc. Main conjectures (theorems in many cases) in Iwasawa theory predict that these invariants are intimately related to the growth of class groups and Selmer groups along certain towers of number fields. For example, it can be shown that the p-adic L-function associated to a certain Dirichlet character of modulus p=37 has one root on the open unit disc. The classical Iwasawa main conjecture (a theorem of Mazur-Wiles) then implies that 37 divides the class number of all fields in the tower Q(w_{37^n}), extending the result of Kummer discussed above.
The p-adic L-functions encountered in the Iwasawa theory of modular forms do not always have finitely many roots, thus their lambda-invariants may not be well-defined. This complication can be avoided by restricting to ordinary modular forms (those whose pth Fourier coefficients are p-adic units), but for nonordinary modular forms it is conjectured (and known in many cases) that one always has infinitely many roots. Amazingly, even in the nonordinary setting there are ways (see here, here, or here, for example) to get at a finite number of 'interesting roots' and one can then ask about their meaning in the context of a main conjecture as above.