_{[Edit: I never intended or expected this answer to be voted so high, so I didn't bother stating up-front the bleeding obvious: that if you suffer from the delusion that The Elements of Style is a book of grammar rules rather than a book about two guys' preferences, well, you're wrong. For a tedious list of rules in the book that aren't grammar rules, see (thanks to Robert Bixler's answer) this article by Pullum, which is a slightly better version of that silly rant. Of course I strongly disagree with the tone of the essay, its false claims about the book, its idea that the faults of the American education system can be blamed on a single book, and its insinuations about the authors' character (seriously!), for which see the rest of this answer.]}

[Preliminary note: the first edition (1918) by Strunk is available online, so you can read the book and judge for yourselves. It's a short book, with mostly sound advice, worth reading once. It's terse and concise.]

Frankly, there's not all that much wrong with the book. And except for a few people like Pullum (the linguist and author of that rant), the book is still beloved by most, and is recommended by writers across the spectrum from Stephen King to Steven Pinker. But it is important to understand what the book is and isn't.

Most importantly, it's a style guide, not a grammar textbook or reference. The authors do not claim to teach grammar; the book should not be used to learn grammar. They make absolutely no claims about English syntax, but at best about what sounds like good style to them. The very fact that some issue is a matter of style implies that there are multiple alternatives that are perfectly grammatical and in use, even by great writers. It is the job of the authors of a style guide to recommend one alternative over others; the book is supposed to be opinionated. Indeed, White's introduction to the book even says

I treasure The Elements of Style for [...] the audacity and self-confidence of its author. [...] He had a number of likes and dislikes that were almost as whimsical as the choice of a necktie, yet...

So of course the book is idiosyncratic; that's one of the endearing things about it. Pullum seems to have confused it for a book on grammar — odd, since he wrote an excellent grammar book and knows what one looks like.

"Rules"

With a book like this, it's implicit that you, the reader, must consider each element of style advice as the opinion of the author, not follow it blindly, and use your own good sense and examples from good writing. If you find that your considered opinion on something they advocate is different, you just ignore the advice and move on — at least it has made you think. The function of the style guide is to draw your attention to these matters of style, so whenever you are about to use one of the constructions mentioned, you can consider both what you wrote and the alternative, and pick whatever sounds better. (So hypothetically, even a book that inverted all the rules would still be quite useful, since it would draw attention to the same matters.) Accordingly, Strunk and White freely violate their advice whenever an alternative is better, as all good writers do, and as the reader is expected to do. Instead of seeing them as setting an example, Pullum calls the authors "hypocritical" for "flaunting the fact that the rules don't apply to them".

[More generally, his unjustified annoyance with the book seems to arise from his rabid intolerance for anyone stating a "rule" based on their preference unless it's already a rule of grammar... in which case they wouldn't need to state the rule at all, would they? :-) Elsewhere, he has written another rant on George Orwell's famous essay Politics and the English Language, again over-obsessing on just one of Orwell's rules, his one against clichés: "Never use a metaphor, simile, or other figure of speech which you are used to seeing in print." He needlessly points out the obvious fact that this is often infeasible and that Orwell violated his rule, and calls the essay "dishonest and stupid".]

"Limp platitudes" and vacuous advice?

The majority of the book, and certainly the most memorable part, deals with those vague little things that make up "style": "Use definite, specific, concrete language", "Omit needless words", "Keep related words together", "Do not overstate", and, most ineffable of all, "Be clear". These are all followed by little explanations that are generally good, but ultimately, you just have to keep the principles in mind. These "limp platitudes" (so called by Pullum) have in fact been the most useful parts of the book to most writers; even one of the writers allegedly criticizing the book (though I'll note that besides Pullum they all say it's a useful book, just misapplied and overrated) says:

Any young person prone to getting tattoos might consider having a few of these permanently engraved where they can readily be seen: Omit needless words. Use concrete language. Be clear. Avoid fancy words. Revise and rewrite. Pure gold.

Now for a few specific points from Pullum's rant:

"Use the active voice"

Strunk & White advocate using the active voice. This obviously doesn't mean anything as silly as the idea that they want to rid the English language of the passive voice. The passive voice is often a better choice as S&W say themselves, but the point they make — that the active voice is generally more direct and vigorous, and that direct and vigorous writing is generally preferable — is true. In fact, if you look, their advice is quite qualified: here's a passage from Pullum's rant itself:

After this unpromising start, there is some fairly sensible style advice: The authors explicitly say they do not mean "that the writer should entirely discard the passive voice," which is "frequently convenient and sometimes necessary." They give good examples to show that the choice between active and passive may depend on the topic under discussion.

Indeed they do. Pullum then goes to falsely claim that they misidentify the passive voice three out of four times. But they do nothing of the sort; what they give are not examples of passive usage, but examples of how transitive verbs in the active voice can be used:

Many a tame sentence of description or exposition can be made lively and emphatic by substituting a transitive in the active voice for some such perfunctory expression as there is, or could be heard.

There were a great number of dead leaves lying on the ground. --> Dead leaves covered the ground.

The sound of the falls could still be heard. --> The sound of the falls still reached our ears.

The reason that he left college was that his health became impaired. --> Failing health compelled him to leave college.

It was not long before he was very sorry that he had said what he had. --> He soon repented his words.

Here, as they say, they are giving (good) examples of how active transitive verbs can make sentences more forceful (or Strunk's strangely preferred "forcible"). This has nothing to do with the grammatical categories of active and passive voice of course, but as this is not a grammar textbook, and as stylistically this is the same thing of making sentences more direct, this is the logical place to put it in this famously terse book. It should be obvious that they aren't giving four examples of converting a sentence from passive voice to active voice. (Why would you expect them to do such a thing? Is this a grammar textbook for children? It is obvious that they assume the reader knows what the active and passive voice are; it probably never occurred to Strunk that someone who does not know may read his "little book".) The structure of the section is clear: first they say why the active voice is often preferable, then they show when it is not, then they move on to other related matters. They wouldn't have returned to harp on the first point again.

"None of us", "that-which"

Pullum claims that the authors claim that "None of us are perfect" is a grammar mistake. Of course, since S&W is not a grammar book and its authors do not claim that anything in the book is a grammatical mistake, Pullum's claim is trivially false. (Indeed their only pronouncements explicitly on grammar seem to be to say that something is grammatical: they say of "A group of us taxpayers protested" that "The wording, although grammatically defensible, is rarely apt", and elsewhere, "There is nothing wrong with the grammar...") But I'll go further and note this on their section about "none":

A plural verb is commonly used when none suggests more than one thing or person.

 None are so fallible as those who are sure they're right.

(This is kind of tautologous anyway; none suggests a plural when used with "are".)

Similarly, of "which" and "that" — a tiny note in a section on misused words — what they say is (bolding mine):

The use of which for that is common in written and spoken language ("Let us now go even unto Bethlehem, and see this thing which is come to pass."). Occasionally which seems preferable to that, as in the sentence from the Bible. But it would be a convenience to all if these two pronouns were used with precision.

So you see they're expressing a view (idiosyncratic, if you like) not on right and wrong, but on what would be "convenient to all" because of "precision" (a dream of keeping words separate, against the tide of usage). Showing, as Pullum does, that "which" is common for restrictive clauses in written and spoken language only reiterates what they already said. (The breed of anti-prescriptivist linguists from which Pullum comes seems to hold that no one ought to express a preference about language, just accept established usage.)

Conclusion

Admittedly, parts of the book are outdated — some of what sounded better in 1918 sounds worse in 2010, and the book hasn't completely caught up. Ideally, another book dispensing the kind of style advice for which S&W is noted would take its place, providing the kind of advice on composition that goes beyond mere grammar or Chicago Manual of Style advice — but none of S&W's critics appear to have made recommendations (that I'm aware of); the book is still useful. Now, there is a lot of valid criticism to be made about the book being handed out with "follow these [blindly]", its tips being treated as inviolable rules, producing "nervous cluelessness" in students about grammar, about Microsoft Word highlighting every use of the passive voice or restrictive which as a grammar mistake, and so on. Apparently, in the United States, this quirky little book has become "the primary vehicle through which grammar was taught to college students and presented to the general public". While I recoil in horror at such lunacy (if such a thing is indeed true), all of these seem failures of the education system to me. Blaming such misuse on the book itself, and calling its authors an assortment of names ("bumblers", "incompetents", etc.) is just mean-spirited.

Small children have a particular writing style that teachers often mark as wrong.

We had a field trip. And we went to the zoo. And we saw monkeys. And they were funny. And then we went home. And the bus was noisy.

Nobody thinks that's a well-written story. So the teacher circles all the "And"s and says "don't start a sentence with and". But somehow we all internalize that as a rule for all of life — which it isn't.