I used to say we look at why a given society or set of societies punishes in the way(s) it does. We might further add that our credo is examining the social, political, and economic factors---that is, things other than crime rates---that influence punishment. The other definition I can give is more vague and doesn't really say much about the field: we study punishment as a social institution or we study punishment from social science and humanistic perspectives. I also gave this definition to my colleague, but they did not find it useful, and I would agree---it says what we study, and some basic information about how we study it, but not really why we study it, what we're after.
Three caveats before moving on: I say crime rates, not perceptions of crime rates, because we know perceptions of crime rates are really important and that perceptions often have little to do with the actual rate of crime (e.g., Beckett 1997).
The second caveat is, at least as I (and a lot of other folks, but maybe not everyone) see it, punishment should be broadly construed. It doesn't have to be state authorized punishments---it can be things that in years past people called social control (but that has become unpopular recently for reasons I'm not yet aware of). In fact, there has been an explosion over the last ten or so years, really going back to about 2010, of people exploring non-punishment punishments---that is, things that many wouldn't call punishment because they are not formally/officially punishment but they sure feel like punishment to the people experiencing them (e.g., Beckett and Herbert 2010, Lynch and Hannah-Moffat 2011, Zedner 2015, Super 2017).
The final caveat is I'm defining the field intuitively rather than based on what's published in what journal. Probably most people who self-identify as P&S scholars publish in P&S, but not everyone who publishes in P&S self-identifies as a P&S scholar. Likewise, self-identified P&S scholars publish in a variety of other journals, including various disciplinary journals (e.g., ASR/AJS, ANNALs, Criminology, BJC, ANZJOC, TC) and interdisciplinary journals (LSR, LSI), among many others.
But in thinking about this intuitively, I'm also prioritizing my unique training, sets of preferences, and network. People in other countries, or in the same country but trained in a different department, or with different research interests may have different takes. Interestingly, there are a lot of people I would label as P&S scholars who don't seem to self-identify that way. (It's an interesting question to think about---how might you empirically define a field. L&S scholars writing review articles trying to identify the L&S "canon" have sometimes used surveys, syllabi, LSR articles, the most-cited articles. Others have suggested we should look at conference presentations. Still others have pointed out that most of these methods are pretty American-centric approaches. Methodologically and theoretically, it's a really interesting question, I think anyway.)
Getting back to these definitions.... Although my preferred definition (abbreviated as "what social factors cause punishment") sounds very macro, there is a greater variety in it than may at first appear. While some of the earliest studies were pitched at the macro level (Durkheim, Marxists, Foucault, Garland), there has been a lot of work at the micro level that still falls under this definition. For example, people who study the behavior of contemporary courts (Van Cleve, Koehler-Haussman) or policing behavior (Stuart) are looking at the micro-level, situational, and interactional factors, embedded in larger frameworks of politics, racial and gender hierarchies, and poverty, that shape penal policies and the experience of punishment---that is, the causes of punishment (and punishment-like experiences, even if they aren't formally punishment).
What gets left out?
But despite its flexibility, I'm not sure my definition (or perhaps characterization) holds anymore. A lot of P&S scholars study other things that are sometimes termed critical criminology and prison sociology. For example, the lineage that begins with Clemmer and Sykes and continues on to a number of scholars today doing prison ethnographies to understand the prison experience, mostly within a sociological tradition (or some would say criminological) tradition. It certainly seems like an affiliated subject and there is a lot of crossover (many of us do both P&S and prison sociology, if we're to refer to them separately, which perhaps we shouldn't). But particularly when looking at prison ethnographies as prisoners' reactions to their confinement (how I usually describe my work, for example) this seems a step removed from the question of what causes punishment. However, if we look at prisoners themselves as one of the causes of punishment (which takes more careful wording than I can muster without implying that prisoners are somehow responsible for their situation--not what I want to imply), along with their organizational setting and the role of the criminal justice actors and policies, then it would still fit. It's not so much about explaining formal policies of punishment and more about explaining the experience of punishment.
There might be even greater overlap with critical criminology in terms of the overall project of understanding penal trends and developments, if sometimes from a different epistemic or philosophical bent. However, despite the overlap, there seem to be a number of folks who identify as P&S scholars who would not self-identify as critical criminologists, and vice versa.
One area that is very likely (and problematically) excluded from this definition is one subset of what might be called the sociology of punishment. In this category, I'm thinking about the many scholars studying the consequences of punishment. (Of course, "sociology of punishment" is also too narrow as there are several notable political scientist studying the political ramifications of our penal trends for democracy generally and voting specifically. However, I've not heard anyone refer to the political science of punishment, while the politics of punishment seems to transcend the soc/poli sci boundary.) This area became really popular in the early 2000s with scholars looking at the consequences of mass incarceration and its associated policies on exacerbating inequality, particularly racial stratification, along class, education, income, employment, health, voting, and other basic facets of life (e.g., Pager 2003, Pettit and Western 2004, Wakefield and Uggen 2010). While you could argue this work has downstream consequences that increase people's chances of getting scooped up in the CJS net, the primary focus is on the consequences of punishment rather than the causes. I find this research super interesting, and don't want to exclude it, but it's an example of one of the limitations of my preferred definition or characterization of the field.
Finally, I want to mention a set of research that predates the rise of P&S as a recognizable area, which I do date back to the founding of P&S in 1999. That's the work by Stan Cohen in 1979 and 1985 on net widening and mesh thinning and the failure of progressive penal reform generally. This strikes me very much as a P&S project, but most folks working in this tradition (at least in the US) were in criminology departments. This is also around the same time as some early prison histories---Rothman (1971 and 1980), Ignatieff (1978), Foucault (1965 and 1977), Melossi and Pavarini (1981)---that were also pretty skeptical about reform. Interestingly, I'd put all of this in the P&S category---it's all about what causes punishment---in this case what causes penal reforms to fail, how penal reforms play out in practice, why punishment turns punitive despite apparent efforts to the contrary, all more specific versions of why a society punishes as it does. But these works all came out before we had something called "punishment and society."
Returning to the Field
Although I've not done a systematic review, it does seem like P&S the journal is becoming more big tent---this is a general trajectory of a lot of journals over time. For example, look at the diversity articles in LSR. This is not to say that either is as diverse as it could be---there are certain major gaps in theories, methods, topics, groups, and countries represented. But relative to their earlier days, it seems they are more diverse.
Consequently, again, it might be the case that the definition I used to give is too narrow to capture what might be better called the field of punishment studies---basically people studying punishment from various social science and humanistic perspectives. (To be more inclusive, I've started using this phrase, punishment studies, but it's still exclusive because of the focus on punishment, which people might interpret too literally as excluding, policing, legalization/criminalization, and social control more generally.) Of course, the (other) problem with this label and definition is it really doesn't get at what we do, what our goals are. It just says what we study, but not why. But perhaps in this larger, more developed field, there is no single, central research question---that might have been possible when it was a really small field, but maybe it's not any more, and perhaps a single underlying, broad research question is not desirable anyway.
Before ending, I do want to make clear that my goal here, in writing this post, is not to say who is in or who is out---my default reaction is we should always aim for a big tent. Instead, definitions are helpful; they are often imperfect and sound more objective than the blurry reality, but they are still informative, especially when used with a sense of caution.
So how do you describe punishment and society to your non-punishment colleagues? What's a broad definition or characterization that fits people who self-identify as punishment and society scholars? (And what's a better word than punishment that's more inclusive of people who study policing, regulations, non-punishment punishments, immigration, and other things that we do study and can learn from?) If your or your friends study punishment, but don't identify as a P&S scholar, why don't you/they?