Where do photons go when they are absorbed? The answer I usually get (and I'm paraphrasing here) is that they disappear and are instead absorbed as heat energy.
But I find it hard to believe that the photon simply "disappears." Common sense tells me it must turn into something or other, not just simply poof out of existence; then again, common sense has betrayed me before.
Forgive me if this is obvious; high school physics student here who's just learned about light and is greatly confused by all this.
 A: How are photons created?
An accelerating charged particle generates photons tangentially as well as a decelarating one. Where do these photons come from? From the energy carried by the electron. In this sense photons are just a packet of energy which is associated with the electromagnetic field. This type of interactions of electrons and ions with fields happens in the photsphere of the sun, for example, generating the light spectrum we observe.
A photon can interact with charged particles and give up part of its energy or even all of it, and then it "disappears".
Photons can also be produced when electrons that are bound in atoms by the electric field of the nucleus , in steady orbitals but in an excited energy level, fall to the lower energy level releasing a photon. A photon of the same energy will be able to kick the electron to the higher energy level, disappearing in the process.
This appearing and disappearing is not an attribute of photons only. In general particles meeting  their antiparticles disappear, because all quantum numbers add up to zero. An electron meeting a positron disappears into two photons. Where do the electron and positrons go? The photon is a simpler particle as it has less quantum numbers to conserve, but the phenomenon exists for all particles in special situations, which you will find out if you carry on to study physics.
A: Well, the answer you usually get is half right. They do disappear (more on this in a second). I'd hesitate to say they turn into "heat energy," both because we don't use the term "heat" that way in a technical sense and because most of the time we like to talk about atoms absorbing photons. In this case the energy of the photon becomes potential energy of the electron that made the transition, and there's no need to talk about heat.
Now, can the photon disappear? The short answer is yes. When you talk about things "not simply poofing out of existence" what you're really describing is like a conservation law. For instance, we say that energy is neither created nor destroyed. Your intuition that things aren't just "poofed" out of existence is probably due to your everyday experience that objects generally can be broken into parts, but not usually destroyed. This isn't true in the particle physics sense, usually. The energy carried by that photon has to be accounted for, as does its momentum and angular momentum. But "photon number" is not a conserved quantity the way that energy or (for instance) electric charge are. A photon really is just a way of looking at disturbances/excitations in the electric field, and so its "destruction" just represents that energy that was present in the field has been moved into some other mode. 
A: A photon, unlike some other particles, has no number that must be conserved, thus when absorbed all of the energy present goes into exciting the particle which absorbed it, allowing no laws to be broken. This is due to Noether's Theorem. http://en.m.wikipedia.org/wiki/Noether%27s_theorem
A: When you turn on a lightbulb, you easily create many photons.  They can go away just as easily. That's because they are bosons and they have no charge. 
Think of waves on a pond. Where do they "come from" when you throw a stone in? Where do they go when they dissipate?
That's actually a very good analogy in some ways because the math that describes transverse waves is the same, but different in a very fundamental way: the waves are quantized.
In quantum field theory, the field (the pond surface) is everywhere, and it may become excited (throwing a stone in). It's the additional step of "all or nothing" that gives you particles, but that's another step added on top of the issue. Where does a lump in a hall carpet go if you manage to stomp it out rather then shift it? The lump is not a "thing" but a "state". (I'm reminded of "where does your lap go when you stand up?". It is funny in ascribing thing-ness in the same way as an object, but it is a description of a state, not an atom of matter.)
A: I am going to answer the question in the simplest terms I know.
Assume you have only one hydrogen atom.  If a photon of the "correct" frequency "hits" the atom, its electron will jump to a higher excited state.  What this means, is that the energy of the photon (the photon itself, since it has no mass) is used (goes into) to make the electron move to a higher excited state. In other words, the kinetic energy of the photon is converted to potential energy (the electron has higher potential energy in the higher excited state).  
A: Once emitted photons are indivisible units. When a photon hits an electron,both moving in the same direction, the photon will be partially absorbed and the electron emits another photon with lower energy. This happens for example at linear particle accelerators. The energy from the photon partially goes over to the electron and the electron moves faster. If a photon hits the electron, both moving against each other, the electron gets decelerated and emits photon(s) with in sum more energy than the incoming photon has.
There is a chance to imagine how the photon goes over to the electron. For this one have to think about how an electric field - and a magnetic field too - can be quantized. For this one need two different quanta, one of them is on the end of the electrons and the antiprotons electric fields and one quanta, which is on the end of the positrons and the protons electric fields. I see clearly the objection of well educated people. Physics claims the infinity of the electric field of electrons and the other charged particles.
The postulation of such two quanta has some charme. From this quanta it's easy to design electric and magnetic field lines. And the most important point is that all the photons (of different energy) are made from this two quanta too.
Photons always are composed from the equal number of both quanta. Negative charged particles have more of the negative quanta and this difference is equal for all electrons and antiprotons. The same difference but with majority for positive quanta positrons and protons have. Emitted and absorbed  photons don't change the particles charge. But the quanta from the photons will be stored partially on the charged particles or these particles gave the quanta back in form of photons of lower energy.
This concept allows to show, that accelerated particles have higher mass and their charge get shilded more and more. And this concept - under the presumption that field lines exist and this field lines are made from this two quanta in clusters with continuous changing numbers - allows to show that the attraction of the electron and the proton in the atom has discrete limits.
And now please forget about this two quanta. They are only a Gedankenexperiment because until now the electromagnetic spectrum seems to be continuous and there is no evidence for quantisation. But it's an amazing imagination to see how photons do not disappear when they hit a electron but travel biggyback on charged particles.
I hope not to be sunk.
