A single photon can easily be detected by a photomultiplier. The basic idea is that a photon hitting a metal plate in the tube ejects an electron from the metal plate by the photoelectric effect. An electric field inside the photomultiplier then accelerates the electron until it slams into another metal plate, releasing a bunch of electrons. These are then accelerated to a third metal plate, etc. The end result is a sizable current we can measure. The Wikipedia article is quite good and has more detail. You might be able to build a crude one at home with a lot of dedication, but it's a delicate device requiring a vacuum and quality electronics. These devices work for IR to UV light.
We can also measure individual photons with a scintillation counter. I used these in a couple of undergraduate labs to detect x-ray radiation from nuclear processes. They work by detecting when a photon (usually high-energy) ionizes an atom in some particular substrate, so they're tuned to detect photons at certain ranges of frequencies. The scintillator does not directly detect these photons, but converts them to several lower-energy photons that we can detect with other means to infer the high-energy photons' presence, so you'll need some more electronics to go with it. Still, we were able to watch single-photon events get counted in lab. (Thanks dmckee for clarification in comments).
The "rod" photoreceptors in your eye may be able to detect single photons. So you can detect single photons at home without any equipment, under the right circumstances.
One device you can fairly easily build at home is a cloud chamber, but it will detect mostly $\alpha$ and $\beta$ radiation rather than photons. However, you might see trails from high-energy photons (gamma radiation). There should be lots of sets of instructions on the web for how to build one.