In the end nearly all life on Earth depends on sunlight for its energy source. Heterotrophs like ourselves are a step or two away from the sunlight, but ultimately it's the process of photosynthesis that fuels our food and thus ourselves. Photosynthesis also, as a byproduct, provides the air we breathe. The three experts who talked about it on In Our Time were Nick Lane (University College London), Sandra Knapp (Natural History Museum) and John Allen (Queen Mary, University of London).
Photosynthesis happens in plants, in structures called chloroplasts inside plant cells. At the botanical level Knapp explained that photosynthesis is the plant taking in CO2 and water, and turning those into oxygen and sugars using the energy from sunlight. After she had set the scene the two biochemists moved on to talk a bit about the molecules involved in making this process work. Lane explained the complexity of the protein complexes that are needed, in terms of their size. On the one had they're very small - each chloroplast is less than a tenth of a millimetre across, yet they are packed full of thousands of clusters of photosynthetic apparatus. But from another perspective they're very big - if you were to be shrunk to the size of an oxygen atom then the photosynthetic complexes would look like vast industrial cities.
Chlorophyll is a critical component of the process, it does the actual light harvesting. Allen gave quite a good verbal description of a chlorophyll molecule - first imagine a line of four carbon atoms with a nitrogen at the end, and then imagine this formed into a ring (a pentagon). Then imagine four of those in a line, then form them into a ring with the nitrogens all in the centre. This is the head of the chlorophyll molecule, and in the centre spot sits a magnesium atom which is essential to the process. One of the four rings has a tail which is insoluble in water but soluble in fats, and so it is what anchors the chlorophyll in the chloroplast membrane (which is made up of fats).
There are actually 300 or so chlorophyll molecules involved in each photosynthetic operation. The light is absorbed by one and then an electron is excited and leaves the chlorophyll molecule. This bumps into another and detaches an electron from it, and it bumps around a bit like a ball in a pinball table. Eventually it reaches one special chlorophyll molecules which uses the energy to "crack" a water atom. They didn't really explain the details of the process (and I have long since forgotten them from the days when I had it memorised!), but from here you either follow a process that ends up with glucose (stored energy) and oxygen (toxic waste) or you use the protons from the water (protons are hydrogen nuclei, so that's what you get if you break up the water and strip the electron off the hydrogen atom) to generate ATP (the energy currency of all cells).
One of the biochemists (Lane, I think) said that ATP is a bit like the coins you put in a coin operated machine. Any time a cell needs energy to do something it uses ATP to power the process. ATP is made either during photosynthesis, or by a process called respiration that both plants and animals do. In essence both do the same thing, but photosynthesis starts with light and respiration starts with glucose. Both processes build up a proton gradient across a membrane - on one side of the membrane there are lots of protons (generated from the H2O or glucose), on one side there are very few. Movement of these protons through the membrane only occurs in channels created by protein complexes that use the energy of this potential difference to generate ATP.
So that's, in basic terms, the process. They also talked a bit about why plants are green - which is one of those "that's a good question, but we're not sure" moments. In one sense (which Knapp pointed out) chlorophyll is green because that's the wavelength of light it reflects. But more interesting is why plants aren't black - surely it would be most efficient to absorb all the light? There is some idea that higher wavelengths of light might damage the plants, so are reflected, but it's not that simple because they don't just absorb red light (which would be safest). In this bit of the discussion they also mentioned the nifty reason why rainforest plants tend to have red undersides to their leaves. Not much light makes it down to the leaves of plants that aren't up in the canopy, so it's important to get as much as you can out of the light you do get. So the red underside reflects red light back up to the top surface of the leaf for another chance of using the energy from it.
Another subject covered was the evolution of photosynthesis, and how plants acquired the ability. And what effect this had on the planet. Photosynthesis evolved in cyanobacteria, which are single celled organisms. Chloroplasts are actually descendants of these free living organisms, which were absorbed or engulfed by ancestral plant cells. So plants didn't evolve the ability to photosynthesise themselves, instead they make use of a cell that already had the ability. The evolution of photosynthesis had a huge effect on the planet - using up the CO2 in the atmosphere had a cooling effect, and actually led to an ice age, a snowball earth. Release of O2 was also not good - it is very reactive, and was actually toxic to most organisms at the time. Bragg was fascinated by the idea that something that's so essential to most life nowadays was once a toxic waste product.
This was a bit of an odd subject for me to write up - I think I've mostly covered what they talked about, but I'm very aware that I used to know a lot more about it. I can remember having the photosynthetic pathway memorised (along with the various steps in respiration too), I just can't remember any of it! So I know the above is simplified, but I no longer know what the details should be. A bit of a weird sensation.