Wow, this one has been over-answered already, I know... but it is such a fun question! So, here's an answer that hasn't been, um, "touched" on yet... :)
You sir, whatever your age may be (anyone with kids will know what I mean), have asked for an answer to one of the deepest questions of quantum mechanics. In the quantum physics dialect of High Nerdese, your question boils down to this: Why do half-integer spin particles exhibit Pauli exclusion - that is, why do they refuse to the be in the same state, including the same location in space, at the same time?
You are quite correct that matter as a whole is mostly space. However, the specific example of bound atoms is arguably not so much an example of touching as it is of bonding. It would be the equivalent of a 10 year old son not just poking his 12 year old sister, but of poking her with superglue on his hand, which is a considerably more drastic offense that I don't think anyone would be much amused by.
Touching, in contrast, means that you have to push - that is, exert some real energy - into making the two objects contact each other. And characteristically, after that push, the two object remain separate (in most cases) and even bound back a bit after the contact is made.
So, I think one can argue that the real question behind "what is touching?" is "why do solid objects not want to be compressed when you try to push them together?" If that were not the case, the whole concept of touching sort of falls apart. We would all become at best ghostly entities who cannot make contact with each other, a bit like Chihiro as she tries to push Haku away during their second meeting in Spirited Away.
Now with that as the sharpened version of the query, why do objects such a people not just zip right through each other when they meet, especially since they are (as noted) almost entirely made of empty space?
Now the reflex answer - and it's not a bad one - is likely to be electrical charge. That's because we all know that atoms are positive nuclei surrounded by negatively charged electrons, and that negative charges repel. So, stated that way, it's perhaps not too surprising that when the outer "edges" of these rather fuzzy atoms get too close that their respective sets of electrons would get close enough to repel each other. So by this answer, "touching" would simply be a matter of atoms getting so close to each other that their negatively charged clouds of electrons start bumping into each other. This repulsion requires force to over come, so the the two objects "touch" - reversibly compress each other without merging - through the electric fields that surround the electrons of their atoms.
This sounds awfully right, and it even is right... to a limited degree.
Here's one way to think of the issue: If charge was the only issue involved, then why do some atoms have exactly the opposite reaction when their electron clouds are pushed close to each other? For example, if you push sodium atoms close to chlorine atoms, what you get is the two atoms leaping to embrace each other more closely, with a resulting release of energy that at larger scales is often described by words such as "BOOM!" So clearly something more than just charge repulsion is going on here, since at least some combinations of electrons around atoms like to nuzzle up much closer to each other instead of farther away.
What then guarantees that two molecules will come up to each other and instead say "Howdy, nice day... but, er, could you please back off a bit, it's getting stuffy?"
That general resistance to getting too close turns out to result not so much from electrical charge (which does still plays a role), but rather from the Pauli exclusion effect I mentioned earlier. Pauli exclusion is often skipped over in starting texts on chemistry, which may be why issues such a what touching means are also often left dangling a bit. Without Pauli exclusion, touching - the ability of two large objects to make contact without merging or joining - will always remain a bit mysterious.
So what is Pauli exclusion? It's just this: Very small, very simple particles that spin (rotate) in a very peculiar way always, always insist on being different in some way, sort of like kids in large families where everyone wants their unique role or ability or distinction. But particles, unlike people, are very simple things, so they only have a very limited set of options to choose from. When they run out of those simple options, they have only one option left: they need their own bit of space, apart from any other particle. They will then defend that bit of space very fiercely indeed. It is that defense of their own space that leads large collections of electrons to insist on taking up more an more overall space, as each tiny electron carves out its own unique and fiercely defended bit of turf.
Particles that have this peculiar type of spin are called fermions, and ordinary matter is made of three main types of fermions: Protons, neutrons, and electrons. For the electrons there is only one identifying feature that distinguishes them from each other, and that is how they spin: counterclockwise (called "up") or clockwise (called "down"). You'd think they'd have other options, but that too is a deep mystery of physics: Very small objects are so limited in the information they carry that they can't even have more than directions to choose from when spinning around.
However, that one options is very important for understanding that issue of bonding that must be dealt with before atoms can engage in touching. Two electrons with opposite spins, or with spins that can be made opposite of each other by turning atoms around the right way, do not repel each other: They attract. In fact, they attract so much that they are an important part of that "BOOM!" I mentioned earlier for sodium and chlorine, both of which have lonely electrons without spin partners, waiting. There are other factors on how energetic the boom is, but the point is that until electrons have formed such nice, neat pairs, they don't have as much need to occupy space.
Once the bonding has happened, however - once the atoms are in arrangements that don't leave unhappy electrons sitting around wanting to engage in close bonds - then the territorial aspect of electrons comes to the forefront: They begin defending their turf fiercely.
This defense of turf first shows itself in the ways electrons orbit around atoms, since even there the electrons insist on carving out their own unique and physically separate orbits, after that first pairing of two electrons is resolved. As you can imagine, trying to orbit around an atom while at the same time trying very hard to stay away from other electron pairs can lead to some pretty complicated geometries. And that too is a very good thing, be cause those complicated geometries lead to something called chemistry, where different numbers of electrons can exhibit very different properties due to new electrons being squeezed out into all sorts of curious and often highly exposed outside orbits.
In metals it gets so bad that the outermost electrons essentially become community children that zip around the entire metal crystal instead of sticking to single atoms. That's why metal carry heat and electricity so well. In fact, when you look at a shiny metallic mirror, you are looking directly at the fastest-moving of these community-wide electrons. It's also why in outer space you have to be very careful about touching two pieces of clean metal to each other, because with all those electrons zipping around, the two pieces may very decide to bond into a single new piece of metal instead of just touching. This effect is called vacuum welding, and it's an example of why you need to be careful about assuming that solids that make contact will always remain separate.
But many materials, such a you and your skin, don't have many of these community electrons, and are instead full of pairs of electrons that are very happy with the situations they already have, thank you. And when these kinds of materials and these kinds of electrons approach, the Pauli exclusion effect takes hold, and the electrons become very defensive of their turf.
The result at out large-scale level is what we call touching: the ability to make contact without easily pushing through or merging, a large-scale sum of all of those individual highly content electrons defending their small bits of turf.
So to end, why do electrons and other fermions want so desperately to their own bits of unique state and space all to themselves? And why, in every experiment ever done, is this resistance to merger always associated with that peculiar kind of spin I mentioned, a form of spin that is so minimal and so odd that it can't quite be described within ordinary three-dimensional space?
We have fantastically effective mathematical models of this effect. It has to do with antisymmetric wave functions. These amazing models are instrumental to things such as the semiconductor industry behind all of our modern electronic devices, as well as chemistry in general and of course research into fundamental physics.
But if you ask the "why" question, that becomes a lot harder. The most honest answer is, I think, "because that is what we see: half-spin particles have antisymmetric wave functions, and that means they defend their spaces."
But linking the two together tightly - something called the spin-statistics problem - has never really been answered in a way that Richard Feynman would have called satisfactory. In fact, he flatly declared more than once that this (and several other items in quantum physics) were still basically mysteries for which we lacked really deep insights into why the universe we know works that way.
And that, sir, is why your question of "what is touching?" touches more deeply on profound mysteries of physics than you may have realized. It's a good question.
Here is a related answer I did for S.E. Chemistry. It touches on many of the same issues, but with more emphasis on why "spin pairing" of electrons allows atoms to share and steal electrons from each other -- that is, it lets them form bonds. It is not a classic textbook explanation of bonding, and I use a lot of informal English words that are not mathematically accurate. But the physics concepts are accurate. My hope is that it can provide a better intuitive feel for the rather remarkable mystery of how an uncharged atom (e.g. chlorine) can overcome the tremendous electrostatic attraction of a neutral atom (e.g. sodium) to steal one or more of its electrons.