The proofs which natural proofs describe show, either directly or indirectly, that a boolean function has a certain natural combinatorial property. Under the assumption that one-way functions exist with "exponential hardness" as specified in their main theorem, Razborov and Rudich show that these proofs cannot separate certain complexity classes. Notably, assuming one-way functions exist, these proofs cannot separate the complexity classes P and NP.
For example, their article states:
A property is natural if it meets the constructivity and largeness conditions defined by Razborov and Rudich. Roughly speaking, the constructivity condition requires that a property be decidable in (quasi-)polynomial time when the 2n-sized truth table of an n-input boolean function is given as input, asymptotically as n increases. This is the same as time singly-exponential in n. Properties that are easy to understand are likely to satisfy this condition, so simple proof techniques will probably not resolve the P vs. NP question. The largeness condition requires that the property hold for a sufficiently large number of the set of all boolean functions.
A property is useful against a complexity class C if every sequence of boolean functions having the property defines a language outside of C.
Razborov and Rudich give a number of examples of lower-bound proofs against classes C smaller than P/poly that can be "naturalized", i.e. converted into natural proofs. An important example treats proofs that the parity problem is not in the class AC0. They give strong evidence that the techniques used in these proofs cannot be extended to show stronger lower bounds. In particular, AC0-natural proofs cannot be useful against AC0[m]
Razborov and Rudich also demonstrate unconditionally that natural proofs cannot prove exponential lower bounds for the discrete logarithm problem.
There is strong current belief that the mechanism of this paper actually blocks lower-bound proofs against the complexity class TC0 of constant-depth, polynomial-sized threshold circuits, which is believed but not proven smaller than P/poly. However, some researchers believe that the Razborov-Rudich limitations are actually good guidance for what a "super-natural" lower-bound proof might involve, such as properties hard or complete for exponential space.