The prevailing view throughout the 19th century was that logic, and the 'scientific method', could explain everything; it was just a matter of time and hard work before they did actually explain everything. At the beginning of that century the French mathemation Pierre-Simon Laplace set the tone:
An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
By the end of the 19th century this process of explaining 'everything' through logic seemed unstoppable, and the German mathematician and philosopher Gottlob Frege was just finishing his grand attempt to derive all of mathematics from logical principles, when Bertrand Russell wrote to him in 1902 with what is now known as 'Russells Paradox' (although it was originally discovered 25 years earlier by the Danish mathematician Georg Cantor).
The paradox concerns the notion of a 'set', first introduced by Georg Cantor. A 'set' is simply any collection of any objects. You can speak of a basket of apples being a 'set of apples'; everything that is red in the world is 'the set of all red things' (provided you had a clear criterion of the quality 'red'); all the real numbers between zero and one (such as 0.0003, 0.3, 0.5, 0.9889) are the 'set of real numbers between 0 and 1' (mathematicians call such a set 'closed' if it actually contains zero and one, and 'open' if it contains all the numbers between zero and one but does not actually contain 0 and 1 themselves). In fact, a 'set' is very similar to a 'concept' - in most writings about mathematical sets, you can mentally substitute the word 'concept' for the word 'set', and everything still makes good sense. Gottlob Frege mentioned above considered 'set' and 'concept' nearly identical.
There is no reason why a set cannot contain other sets (you can certainly have a concept about other concepts). The set of all sets that contain 5 or less members is itself a set - it would include, for instance, my fruit-bowl of four apples. If a set can contain other sets, can it contain itself? The set of all sets that contains 6 or more members would include itself, since there are very many (more than six anyway) sets that contain 6 or more members; therefore the group as whole of sets containing 6 or more members is itself a set, and contains itself - the set of all sets containing 6 or more members.
Russell asked himself about the set consisting of all sets that are not members of themselves - in Cantor's system this is a well-defined set. If you don't accept the paragraph above that any set can belong to itself, then Rusell's set is simply the set of all sets.
Is Russell's set a member of itself?
If it is, then since it is a member of Russell's set, it must be a set that is not a member of itself.
If it is not, then it must be a member of the family of sets that are not members of themselves - namely Russell's set.
So we have a contradiction either way. A more intuitive way of looking at Russell's Paradox is to consider a barber who shaves everyone who does not shave himself, and no one else. If the barber shaves himself, then he doesn't shave himself; if he doesn't shave himself, then he does. Frege's stated his own version of this paradox in terms of concepts, in which case Russell's set is the concept that does not fall under its defining concept.
There are a number of ways that mathematicians and logicians have tried to circumvent this paradox. The most common method is by making a distinction between a 'set' and a 'class'. However, many thinkers (see references below) hold this, and all the other methods, unsatisfactory. I do myself - most responses simply 'cut out' the paradox and all its consequences; you are left with a large amount of useful logic and mathematics, but at the cost of ignoring what Russell's Paradox is actually implying. For instance, a 'class' is simply a collection that you consider a 'set', but then find that you have a paradox similar to above, so you consider this 'set' to in fact be a 'class' (you cannot have classes of classes, like you can have sets of sets), you call the original idea that your class was a set 'naive', and so the paradox is avoided - note that I say 'avoided', not solved.
What I understand from Russell's Paradox is the 'set of all sets', or the 'concept of all concepts' is in some sense too big or too powerful for everyday logic. But surely, if that is the case, there are many other concepts that are also 'too big' in the same way - what about: God, beauty, truth, the purpose of my life?
Throughout the 20th century, mathematicians and logicians built on Russell's paradox, and many results were derived logically showing that logic itself had limits. The most famous is probably Godel's two Incompleteness Theorems. The second one can be stated:
If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.
What more devastating blow can be given to the program that attempts to build up complete understanding of the universe, and ourselves in it, through logic, or consistent axiomatic systems?
I am not anti-logic. In fact I love logic, and what it can do, and the insights it can generate. And as I write in my previous article, I have recently found, through the Focusing process, how useful logic can be even in the meditative process, from whence it is usually banned. All I am stating here is that logic (or 'consistent axiomatic systems') has limits, and there are concepts I can have, things I can think about, that are simply outside logic's domain.
The trick, of course, is to find the balance between logic and reason, and other modes of knowledge - intuition, gut feeling, direct perception, 'just knowing'. I accept that reason will never take me to where I need to go, but it can provide good 'base camps'. I prefer reason or logic to be on my side, I can accept it being neutral, but I certainly never want to go against what reason tells me.
The Road to Reality by Roger Penrose - provides a comprehensive account of our present understanding of the physical universe, and its underlying mathematical theory, but you need to be comfortable with mathematical equations to read it (or else get a buzz out of not being comfortable with them!) The concepts in my article are covered in chapter 16 of this book, and references therein.
The Free-Definition has a good article on Russell's Paradox, and a good set of links to others, including the responses to the paradox.
There are many formulations of Godel's theorems on the internet. Many popular ones are misleading, or full of too much hype. A good one is here.