tl;dr: In competitive selection processes, selection is a two-place word: there's something being selected (a cause), and something it's being selected for (an effect). The phrase group-level gene selection helps dissolve questions and confusion surrounding the less descriptive phrase "group selection".
(Essential note for new readers on reduction: Reality does not seem to keep track of different "levels of organization" and apply different laws at each level; rather, it seems that the patterns we observe at higher levels are statistical consequences of the laws and initial conditions at the lower levels. This is the "reductionist thesis.")
When I first encountered people debating "whether group selection is real", I couldn't see what there was to possibly debate about. I've since realized the debate is mostly a confusion arising from a cognitive misuse of a two-place "selection" relation.
Causes being selected versus effects they're being selected for.
A gene is an example of a Replicating Cause. (So is a meme; postpone discussion here.) A gene has many effects, one of which is that what we call "copies" of it tend to crop up in reality, through various mechanisms that involve cellular and organismal reproduction.
For example, suppose a particular human gene X causes cells containing it to immediately reproduce without bound, i.e. the gene is "cancerous". One effect is that there will soon be many more cells with that gene, hence more copies of the gene. Another effect is that the human organism containing it is liable to die without passing it on, hence fewer copies of the gene (once the dead organism starts to decay). If that's what happens, the gene itself can be considered unfit: all things considered, its various effects eventually lead it to stop existing.
(An individual in the next generation can still "get cancer", though, if a mutation in one produces a new cancerous gene, Y. This is what happens in reality.)
Thus, cancers are examples of where higher-complexity mechanisms trump lower complexity-mechanisms: organism-level gene selection versus cellular-level gene selection. Note that the Replicating Cause being selected is always the gene, but it is being selected for its net effects occurring on various levels.
There is no debating that genes are selected for both cellular and organismal effects. However, notice also that the organismal effect factors through the cellular effect: the organism dies because of the massive cell reproduction (cancer). There is no magic "layer jumping" from the gene to the organism.
In other words, "organismal effect" is a label we use when the mechanism requires us to think about the entire organism to see what happens. It's a complexity marker. (Note that a powerful enough computer would not need this layer distinction. It would simply simulate the whole system, the way reality does, and see the cells gradually form a tumor, and eventually perish as a result.)
There is also no debating that genes can also have effects at the group level, and that these effects could increase or decrease the number of copies of the gene in existence by effecting the group to survive, grow, "reproduce" (seed colonies elsewhere), or annihilate itself. Of course, these effects will all factor through cells and organisms. Calling them "group-level effects" simply refers to our inability to predict them without thinking about the "big picture".
The debate/confusion now dissolves into the following component questions:
- Q: Does significant group-level gene selection occur? I.e., do we need to run simulations large enough to observe group selection effects in order to get accurate predictions?
A: Probably yes. It's easy to imagine how it could happen, and PhilGoetz just posted with some evidence that perhaps we do. Data collection and model development should continue in this area, so we can better understand the balance between group-level and organism-level gene selection mechanisms.
- Q: Is group-level gene selection fundamental?
A: No. To the best of anyone's knowledge, the effects of a gene on a group factor through (operate via) its effects on individuals, and are thus a statistical consequence of the latter.
- Q: Should we treat group-level gene selection as fundamental? I.e., If we need a large-scale model of organismal evolution, should we program it with extra laws that govern the selection of groups?
A: Probably not. The important thing is that we model the individual interactions accurately enough that the resulting complex group interactions will be accurate, too. If we do that, we're done. If we don't, we're unlikely to guess the correct group-level instructions that will cancel out the inaccuracy. (ETA: Though as nhamann points out, some might still think we can do it.)
- Q: Are genes the only Replicating Causes we should treat as fundamental?
A: For now, no. Until we can model chemical reactions well enough to predict thoughts, we'll need to construct models that at least treat memes as fundamentally separate. Contrast: A gene is a molecular-level cause with effects at all higher levels. A meme is an organism-level cause1 with effects at all higher levels.
From the mixture of yes/no and certain/uncertain answers here, you can see how a lot of unnecessary "debate" could occur if two conversing parties were unwittingly trying to answer two different questions. But now, having clarified what's selected versus what's selected for, and what occurs in reality versus what's fundamental in our model...
...is there anything more to ask?
1 When I say a meme, like the belief "Water can put out fire", is organism-level, I mean that our notion of belief is not meant to ask whether a molecule or a cell believes water can put out fire. Beliefs are configurations of brain cells, not states of individual brain cells. The smallest self-replicating unit that contains this configuration is the human organism, so for evolutionary considerations it's an "organism-level" Replicating Cause: we'd need a simulation on a scale that includes organism competition to see its basic effects.