Why do cones require more light than rods? The title says it. In our eyes, why do cones require more light than rods? Like is it because of the energy needed to change the state of the existing atoms, the molecules or what leads to the need of more light to be activated?
 A: This answer will lean pretty heavily on one review article, Imamoto and Shichida, Cone visual pigments (2013) whose authors I occasionally call I&S in this answer.
According to Imamoto and Shichida, this is a complicated question that has been very difficult to solve because the cone proteins have been rather hard to isolate historically. It is known that the basic function of both the rod and cone proteins is the same; they are both “opsins” with the rod-opsins being perversely called rhodopsins (not literally because it sounds like rod-opsin, but apparently the Greek root is for a rose reflecting its color), and the cone opsins having no standard naming conventions except for that—cone opsins. There are tons of different opsins, as you might expect from something that originally evolved before the vertebrates. So you might refer to like “human red opsin, human green opsin, human blue opsin” or so to refer to our three variants, though each of those lies in a family of red opsins etc. shared in many cases with other species, which families are often called by names like LW for “long wavelength” and so forth; see Terakita, The opsins (2005) for some more details there. 
Similarities in how rods and cones work
Anyway, as far as I can tell both rhodopsins and cone opsins have various similarities:


*

*Both consist of seven parallel α-helix ‘columns’ which create a ‘space’ inside them. Inside that space there is a vitamin-A molecule, which has been twiddled to be covalently bonded to a lysine group in one of the columns. In this conformation that vitamin-A moiety is called retinal.

*Retinal is naturally UV-sensitive; one of the things that the columns do is to change the absorption maximum down into the visual region of light. So different conformations of the protein can be tuned to different peak-absorption wavelengths. (This is only mentioned implicitly in Imamoto and Shichida but is explicitly stated in Terakita.) 

*Retinal has a ring structure and a ‘tail’ and in its natural state the tail is bent; the photon absorption twists the tail out to be straight. This causes a conformal change where two columns of the opsin push out from the center, which exposes a binding site for a family of proteins called transducins. Rods and cones use different transducins etc. but the story is apparently similar either way. (Some opsins in other creatures are really funky: they use the straight retinal at rest, and then a photon twists it back.)

*Transducins have a complicated form where there appear to be two molecular halves that are normally fused together into a mostly-inert lump apparently by hydrogen bonds and maybe van der Waals forces. One part of one half is a little guanosine diphosphate (GDP) assembly. When transducins bind to the opsin, it catalyzes the swapping of that GDP with guanosine triphosphate (GTP), which like ATP is in a higher energy state and can later see its extra phosphate removed. But when that GTP is in there, the two halves no longer fit together, which exposes the juicy internals to the rest of the cell. The cell then uses those juicy internals to create an enzyme (cGMP phosphodiesterase) which breaks down a signaling molecule (cGMP) which normally is very common in the cell.

*It turns out that the visual cells are basically always firing randomly; they are hooked up to the biological equivalent of a NOT gate which only starts firing into the ganglion when the adjoining cell stops firing. But this neuronal firing requires this cGMP to keep the cell walls open, and as it disappears and the cell walls close off, it stops firing. 

*A given opsin handles many transducins before returning back to its natural state; this McGill page says 100ish, while Bionumbers says 1000ish. Bionumbers also states that each transducin apparently only activates about one phosphodiesterase, but each of those then kills many cGMP molecules nearby, further amplifying the signal. If I am reading I&S right, they are stating that actually the opsins regenerate not by simply waiting until the retinal collapses back to its former state, but actually it gets torn out from inside the opsin and replaced with another retinal. (I&S: “Photoexcited pigments are eventually dissociated into all-trans-retinal and opsin, which is then reconstituted into pigment by being supplemented with 11-cis-retinal.”)


Reasons for differences
On top of these similarities, there are two reasons that rod cells are more sensitive than cone cells. The first is the boring one: they appear to have more opsins. This is mostly a bit of original speculation on my part; Bionumbers (ibid.) only specifies that there are around 108 rhodopsins in a rod but does not say how many there are in a cone. But rod cells are generally much bigger and have this structure of “stacked disks” inside the cell, containing many layers of detectors which each contain a bunch of rhodopsin that might absorb incident light. So at the simplest level, it looks like more photons need to hit a cone because the interaction probability is simply lower because there are just more layers of rhodopsin to potentially interact with. (It's possible that I’m wrong and cones just have a much higher density of opsins.)
Well then we get to the more interesting question of whether they do the same thing given that one photon is absorbed. And the answer seems to still be “no,” a cone cell needs to be activated by multiple photons stimulating multiple cone opsins, while rod cells might be single-photon-sensitive or else closer to it. Early speculations held that maybe this was because the cone opsins were simply less responsive pigments than rhodopsins, so that they would absorb photons and then do nothing. I&S report that Shichida et al. measured this responsivity for chicken rhodopsin vs. chicken cone opsins in 1994, and falsified that speculation: they found instead that these photosensitivities were quite comparable. So the opsins are about as photosensitive individually to single photons, given that they absorb them. The rod cells might absorb more, but they both do the same thing when they absorb one.
The more interesting reason that I&S identify for the discrepancy is that cone opsins turn out to deactivate much faster than rhodopsin does (I&S: “a few hundred times faster”). So, they detail, it has the time to convert hundreds more transducins to an active state before it relaxes back into its passive state. So the amplification factor is greater simply because the time resolution of rhodopsin is much lower. As one might imagine in amplifying weak signals, this also appears to cause a large amount of noise of rod cells going into their active state even when they are not exposed to any light at all.
That is not the only possibility; as mentioned above there are two amplification stages, one with how many transducins are activated by an opsin, and one with how many cGMPs are killed by a phosphodiesterase. I have not yet seen any discussion about the latter, so that is maybe an open question. I wouldn't expect the phosphodiesterases to have different strengths, but it’s certainly possible.
A: 
Why do cones require more light than rods?

Rods are used to see at night, they must be very sensitive to light and from an evolutionary standpoint color vision (and polarization) are not necessary for the selection process. We have approximately 120 million rods. Use of rods in low light conditions to see is called scotopic vision.
Cones are used to see at higher light levels and do enable color vision, like rods they lack the ability to discriminate based on polarization. We have approximately 6 to 7 million cones, divided into "red" cones (64%), "green" cones (32%), and "blue" cones (2%); blue is the most sensitive. Use of cones during well lit conditions is called photopic vision.
In addition to the individual sensitivities of the rods and cones the brain plays an important role in vision and the preprocessing of the stimulus received from the eye.
When it is neither particularly dark nor light mesopic vision is used. The switching (or dark adaptation) from lighter to darker conditions is called the Purkinje effect.
Rods and cones are distributed within the eye as follows:

Image from: "HyperPhysics - Rod and Cone Density on Retina".
As indicated by the image above color vision is centrally located while night vision is at the periphery. This means that during the day we see better, and in color, by looking directly at an object. At night we see best by not looking directly at an object, we have developed this ability to avoid being blindsided at night.

From: "Polarization Vision - Jan 2014 - DOI: 10.1007/978-1-4614-7320-6_334-4
  In book: Encyclopedia of Computational Neuroscience, Chapter: Polarization Vision, Publisher: Springer, Editors: Dieter Jaeger, Ranu Jung, pp.1-30:
"Polarization vision is the ability of animals to detect the oscillation plane of the electric field vector of light (E-vector) and use it for behavioral responses. This ability is widespread across animal taxa but is particularly prominent within invertebrates, especially arthropods. ... The most prominent source of polarized light is the skylight polarization pattern, which contains information about the position of the sun in the sky and is thus used for navigation purposes. In insects, E-vectors are detected through specialized regions of the compound eyes. Neurons downstream of the involved photoreceptors respond to changes in E-vectors with modulations of their action potential frequency, so that each neuron is tuned to one particular E-vector.

Sources & References: HyperPhysics - Vision Concepts:


*

*The rods are incredibly efficient photoreceptors. More than one thousand times as sensitive as the cones, they can reportedly be triggered by individual photons under optimal conditions. 

*"Just about at the center of the retina is a small depression from 2.5 to 3 mm in diameter known as the yellow spot, or macula. There is a tiny rod-free region about 0.3mm in diameter at its center, the fovea centralis. (In comparison the image of the full Moon on the retina is about 0.2 mm in diameter.) Here the cones are thinner (with diameters of 0.0030mm to 0.0015mm) and more densely packed than anywhere else in the retina. Since the fovea provides the sharpest and most detailed information, the eyeball is continuously moving, so that light from the object of primary interest falls on this region. ...the rods are multiply connected to nerve fibers, and a single such fiber can be activated by any one of about a hundred rods. By contrast, cones in the fovea are individually connected to nerve fibers. The actual perception of a scene is constructed by the eye-brain system in a continuous analysis of the time-varying retinal image." (Hecht)

Image from: "HyperPhysics - The Fovea Centralis".
Our site Photo.SE also has an interesting Q&A about visual perception: "How can knowledge of human perception of color be used in photography?".
A: This is part of what Wikipedia says about rods:

"Rod cells also respond slower to light than cones and the stimuli
they receive are added over roughly 100 milliseconds. While this makes
rods more sensitive to smaller amounts of light, it also means that
their ability to sense temporal changes, such as quickly changing
images, is less accurate than that of cones."


*

*So rods are collecting light for a longer time, thus making them more sensitive but slower. I have not found a value for how long cones add light. This article says:

Psychophysical studies show that cone reaction times are shorter than rod reaction times, however the estimated delays differ substantially among studies. Several studies reported rod-cone latency differences of 60–80 ms

Reaction times in this study of cones and the claims on Wikipedia regarding how long time rods collect light might not really measure the same thing but rods are obviously slower to respond and cones must thus be collecting light for less time.


*Another reason rods require less light, not on the cellular scale but as a collective, is that several rods are connected to the same neuron cell, thus making for more light sensitivity but reducing the angular resolution.


*Rod cells typically all have a diameter of about 2 µm but this is what Wikipedia has to say about cones:

They are typically 40–50 µm long, and their diameter varies from 0.5 to 4.0 µm, being smallest and most tightly packed at the center of the eye at the fovea"

Obviously if some cones have a diameter eight times smaller than others one would expect them to collect about 64 times less light. As the smallest cones have a diameter of one fourth that of the cones we would expect them to collect one sixteenth the amount of light.


*A fourth reason might be that rods contain more pigments than cones, but I do not know why, maybe the pigments used in cone cells are larger so there is not room with so many of them.


*The three different types of cones are sensitive to light in different frequencies, which, depending on how much light is available at their sensitive frequencies might make them more or less sensitive to light. Rods are sensitive to different frequencies that, maybe, can make them more sensitive in dim lighting. This is what Wikipedia says:

Experiments by George Wald and others showed that rods are most sensitive to wavelengths of light around 498 nm (green-blue), and insensitive to wavelengths longer than about 640 nm (red). This is responsible for the Purkinje effect: as intensity dims at twilight, the rods take over, and before color disappears completely, peak sensitivity of vision shifts towards the rods' peak sensitivity (blue-green).

Rods have a peak sensitivity at 498 nm, "blue cones" at 420 nm, "green cones" at 534 nm and "red cones" at 564 nm. This affects their sensitivity depending on the wavelengths of light present, but I do not know if this always benefits the cones.
