I am reading Hawking's "Brief answers". He complained that black holes destroy information (and was trying to find a way to avoid this). What I don't understand: Isn't deleting information quite a normal process? Doesn't burning a written letter or deleting a hard disk accomplish the same?
(The answers by Mark H and B.fox were posted while this one was being written. This answer says the same thing in different words, but I went ahead and posted it anyway because sometimes saying the same thing in different words can be helpful.)
The key is to appreciate the difference between losing information in practice and losing information in principle.
If you write "My password is 12345" on a piece of paper and then burn it, the information might be lost for all practical purposes, but that doesn't mean that the information is lost in principle. To see the difference, compare these two scenarios:
Scenario 1: You write "My password is 12345" on a piece of paper and then burn it.
Scenario 2: You write "My password is ABCDE" on a piece of paper and then burn it.
Exactly what happens in either scenario depends on many details, like the specific arrangement of molecules in the piece of paper and the ink, the specific details of the flame that was used to ignite the paper, the specific arrangement of oxygen molecules in the atmosphere near the burning paper, etc, etc, etc. The variety of possible outcomes is equally vast, with possible outcomes differing from each other in the specific details of which parts of the paper ended up as which pieces of ash, which molecules ended up getting oxidized and drifting in such-and-such a direction, etc, etc, etc. This is why the information is lost in practice.
However, according to the laws of physics as we understand them today, all of the physically possible outcomes in Scenario 1 are different than all of the physically possible outcomes in Scenario 2. There is no way to start with a piece of paper that says "My password is 12345" and end up with precisely the same final state (at the molecular level) as if the piece of paper had said "My password is ABCDE." In this sense, the information is not lost in principle.
In other words, the laws of physics as we understand them today are reversible in principle, even though they are not reversible in practice. This is one of the key ideas behind how the second law of thermodynamics is derived from statistical mechanics.
The black hole information paradox says that our current understanding of physics is necessarily flawed. Either information really is lost in principle when a black hole evaporates, or else spacetime as we know it is only an approximately-valid concept that fails in this case, or else some other equally drastic thing. I think it's important to appreciate that the black hole information paradox is not obvious to most people (certainly not to me, and maybe not to anybody). As a testament to just how non-obvious it is, here are a few review papers mostly written for an audience who already understands both general relativity and quantum field theory:
 Marolf (2017), “The Black Hole information problem: past, present, and future,” http://arxiv.org/abs/1703.02143
 Polchinski (2016), “The Black Hole Information Problem,” http://arxiv.org/abs/1609.04036
 Harlow (2014), “Jerusalem Lectures on Black Holes and Quantum Information,” http://arxiv.org/abs/1409.1231
 Mathur (2011), “What the information paradox is not,” http: //arxiv.org/abs/1108.0302
 Mathur (2009), “The information paradox: A pedagogical intro- duction,” http://arxiv.org/abs/0909.1038
Section 2 in  says:
conventional physics implies the Hawking effect to differ fundamentally from familiar thermal emission from hot objects like stars or burning wood. To explain this difference, ... [technical details]
Section 4.2 in  says:
The burning scrambles any initial information, making it hard to decode, but it is reversible in principle. ... A common initial reaction to Hawking’s claim is that a black hole should be like any other thermal system... But there is a difference: the coal has no horizon. The early photons from the coal are entangled with excitations inside, but the latter can imprint their quantum state onto later outgoing photons. With the black hole, the internal excitations are behind the horizon, and cannot influence the state of later photons.
The point of listing these references/excerpts is simply to say that the paradox is not obvious.
The point of this answer is mainly to say that burning a letter or deleting a hard disk are reversible in principle (no information loss in principle) even though they make the information practically inaccessible, because reconstructing the original message from its ashes (and infrared radiation that has escaped to space, and molecules that have dissipated into the atmosphere, etc, etc, etc) is prohibitively difficult, to say the least.
Note added: A comment by the OP pointed out that the preceding answer neglects to consider the issue of measurement. This is an important issue to address, given that measurement of one observable prevents simultaneous measurement of a mutually non-commuting observable. When we say that the laws of physics as we currently know them are "reversible", we are ignoring the infamous measurement problem of quantum physics, or at least exploiting the freedom to indefinitely defer application of the "projection postulate." Once the after-effects of a measurement event have begun to proliferate into the extended system, the extended system becomes entangled with the measured quantity in a practically irreversible way. (The impossibility of simultaneously measuring non-commuting observables with perfect precision is implicit in this.) It is still reversible in principle, though, in the sense that distinct initial states produce distinct final states — provided we retain the full entangled final state. This is what physicists have in mind when they say that the laws of physics as we currently know them are "reversible." The black hole information paradox takes this into account. The paradox is not resolved by deferring the effects of measurement indefinitely in the black hole case, nor is it resolved by applying the "projection postulate" as soon as we can get away with it in the burning-piece-of-paper case. (Again, the BH info paradox is not obvious, but these things have all been carefully considered, and they don't resolve the paradox.)
Since we don't know how to resolve the measurement problem, either, I suppose we should remain open to the possibility that the black hole information paradox and the measurement problem might be related in some yet-undiscovered way. Such a connection is not currently clear, and it seems unlikely in the light of the AdS/CFT correspondence , which appears to provide a well-defined theory of quantum gravity that is completely reversible in the sense defined above — but in a universe with a negative cosmological constant, unlike the real universe which has a positive cosmological constant . Whether the two mysteries are connected or not, I think it's safe to say that we still have a lot to learn about both of them.
When Dr. Hawking talks about information being destroyed, he is talking about the erasure of all evidence that the information ever existed. In the case of burning a written letter, you could track the trajectory and composition of every smoke particle as the book burned. Since ink and paper generate different kinds of smoke and start at different locations, you could use that information to reconstruct the original book.
Obviously, this is impossible to do in real life, but the information is there to be had. In the same way that energy is neither created nor destroyed, information is neither created nor destroyed. This is why Hawking was suspicious about black holes seeming to retain no evidence of what fell into them.
I must admit, I'm not the most qualified person to answer here.
If you place the burning letter into a sealed box (a closed system) and allow the system to carry itself to its final state, with a perfect model of the physics involved, you can trace the end state back to its original state if you know the properties of the end state perfectly because the constituent particles of the letter carry with them information regarding their previous interactions. That information is preserved in the system, and can be decoded.
If you place a black hole in a box and some in-falling matter, information will be lost behind the horizon, permanently so once the black hole evaporates.