Contrary to claims that the greenhouse effect is complicated, it's really quite simple. The IPCC's definition in the glossary of AR5 WG1 is fine, but can be made even clearer with the help of the fifth figure at http://clim8.stanford.edu/Images/ . All greenhouse gases work this way, there is no need to consider particular absorption lines unless you're using them to estimate no-feedback climate sensitivity for a particular species of a particular molecule, which is extremely difficult to confirm experimentally.
ZeroTheHero suggested I expand on this, so here goes. Though now that I look at jobal6's answer I don't see that my answer below adds much to it.
Longwave radiation in the atmosphere relevant to the greenhouse effect (GHE) divides into three kinds, downwelling, upwelling, and outgoing, respectively DLR, ULR, and OLR.
OLR is what escapes to space. It is 100% responsible for keeping Earth in thermal equilibrium with the Sun, the state in which OLR equals the absorbed solar irradiance (the fraction 1-A not reflected back to space where A is Earth's albedo, nominally 0.3).
Ignoring lensing effects of the kind that causes mirages, all longwave radiation within the atmosphere divides into DLR and ULR according to whether it points to below or above the horizon respectively. (At least that's how I would define it, your mileage may vary.)
ULR is a candidate for becoming OLR, which it does when it isn't absorbed by a greenhouse gas (GHG) molecule or an aerosol particle. Otherwise it either heats the absorbing molecule or particle, thereby returning it to Earth's heat, or is reradiated in a random direction (e.g. stimulated emission?).
DLR differs from ULR only in that none of it can become OLR directly. 100% of DLR is either absorbed by the surface or otherwise behaves in every respect just like the fraction of ULR that doesn't become OLR.
Since that may seem surprising it should be pointed out that although DLR increases with increasing GHGs, as you will read in the many accounts of back radiation, ULR also increases, which people not trained in physics tend to overlook. What matters for heating the surface is not DLR alone but DLR - ULR since the ULR removes heat at the same time DLR is contributing heat.
As a consequence of (i) the Stefan-Boltzmann relation F = σT⁴, (ii) the fact that ULR at any given point originates from a hotter place than DLR at the same point, and (iii) lapse rate maintains a constant temperature difference between any two altitudes, ULR increases faster than DLR, whence the net downward flux DLR - ULR actually decreases with increasing CO2.
So that can't be the reason the surface warms with increasing GHGs. (The IPCC knows this and you won't find anything about back radiation in AR5 WG1. It's also not in Figure 7 of Kiehl & Trenberth 1997, which shows ULR > DLR, frequently overlooked.)
The reason the surface warms is because increasing GHGs make it harder for ULR to become OLR. This traps more heat, which warms not just the surface but the atmosphere and the oceanic mixed layer, OML, enough to increase ULR until the fraction that becomes OLR is restored to the level needed to keep Earth in thermal equilibrium with the Sun.
The thermal inertia of the surface, atmosphere, and especially the OML is enough for thermal equilibrium to take centuries. This is why equilibrium climate sensitivity, ECS, is higher than transient climate response, TCR, which is only allotted 70 years to warm up.
One would also expect the deep ocean to be a further impediment to equilibrium. The tricky part here is that the deep ocean has far more thermal inertia than the OML, while also being better connected to the ice caps than to the OML, namely via the great ocean conveyor belt's thousand-year journey. This makes the ice caps themselves the real further impediment to equilibrium. Basically the deep ocean stays cold, leaving the OML to achieve the appearance of thermal equilibrium.
Eventually the ice caps melt, which I believe is what at least some people mean by Earth System Sensitivity (but I'm a bit vague on that, maybe climate modelers can kick in here). Obviously that equilibrium takes far longer than the equilibrium associated with ECS, there being a heck of a lot of ice to melt.
I think this covers the basic greenhouse effect, GHE. There's a lot more to climate than that but the GHE is the part fundamentally responsible for global warming.
"not the simplistic greenhouse analogy provided for public consumption"
Joseph Fourier is responsible for the analogy. It's a lot better analogy than it's usually given credit for these days. For starters Earth's gravity prevents Earth's atmosphere from convecting to space just like a greenhouse's glass prevents its air from convecting to the environment. And secondly the reason greenhouses need windows that can be opened is that without them the heat trapped by the glass during the day would overheat the plants, which would not be a problem in the case of no greenhouse but completely motionless (non-convecting) air.
The Louvre in Paris has the greenhouse effect problem in spades, both in the main entrance under I.M.Pei's glass pyramid (I've sweated it out there myself, it's horrible on a sunny day) and in the glass-covered Cour Marly inside.
Lastly it should be pointed out that although Earth's atmosphere is thousands of times thicker than a sheet of glass, CO2 at 1000 ppm if chilled to dry ice would cover the Earth with a layer one centimeter thick. So 400 ppm is an excellent approximation to a sphere of glass 4 mm thick suspended somehow over Earth.