One hallmark of the modern research university is the proliferation of Centers and Institutes, groupings of investigators outside the hierarchy of traditional academic departments. I'd like to explain a bit about what these entities are, what (in my view) makes an effective one, and some challenges that these organizations face. There is a great deal of variance across universities with these terms; the version I'm going to describe is mostly what we have at my home institution. Your mileage may vary, and I'd like to hear your thoughts on what makes a great center or institute.
An Institute is an organization that draws members from across different departments (indeed, often from across different Schools such as Natural Science and Engineering), with a strong, usually broad, thematic focus, and with an annual budget for staff and programs that comes largely from internal university funds. Institutes support programs that benefit their membership. Examples of programs include: seminar series; topical workshops and conferences, including interaction with companies, political entities, or the media; visitor programs; educational forays such as interdisciplinary graduate programs, training grants, research experience for undergraduates or teachers, K12 outreach days; endowed postdoctoral fellowships; etc. An Institute is meant to act as a a catalyst or enabling structure to bring together researchers with a common intellectual interest, to foment new and support existing collaborations, and to further research activity in that area. At some universities, an Institute may have its own building or administer shared infrastructure.
A Center is usually a smaller, more focused group of researchers that is often expected to be financially self-supporting through multi-investigator external funding. (Sometimes these are called Laboratories.) Centers often exist within or are founded as the result of institutes. A Center likely concentrates on a portfolio of specific research-related projects, rather than having broad programmatic efforts like an Institute. The US NSF sponsors a number of center programs (MRSECs, ERCs, STCs, CCIs, and formerly NSECs), as does the US DOE (EFRCs).
An effective Center is almost self-defining: It is able to accomplish focused research goals and to raise sufficient external resources to be self-supporting at least on the several year to decade timescale. A good Center is able to identify and adapt to new research opportunities, while realizing which avenues are becoming played-out and should be set aside - basically, effective self-criticism while encouraging creativity through seed projects to generate new activity. Longevity and research productivity metrics are two ways to assess the utility of a Center.
An effective Institute needs to serve its members by successfully supporting and carrying out its programs. Because many Institutes have very broad programmatic goals, this requires serious "buy-in" - a decent fraction of the membership have to be willing to invest their time and energies to ensure that these programs are a success. This only happens when the people involved really believe in the efforts and can see that they and the institution derive real benefit from the work. This means that the Institute has to be responsive to the needs of its membership. At the same time, the university has to assess (via research and funding metrics) the impact of the Institute and its programs, since the university has to decide whether the internal resources of the Institute could have been better spent elsewhere.
Both Institutes and Centers can be vulnerable to budgetary problems (internal and external, respectively) and to lack of engagement by membership. At most places (the University of Chicago seems to be a big exception, since there Institutes have a lot of power) an Institute can be particularly exposed in tight times, since departments and much of their budgets are explicitly necessary for the reaching mission of the university, while Institutes are often viewed as elective or discretionary expenses. In terms of engagement with members, like many organizations, Institutes and Centers succeed by succeeding and fail by failing. You can't force people to collaborate, but once some do arise, productive collaborations lead to further productive collaborations. Overall, Centers and Institutes appear to be key components of successful research universities. It's not clear how these organizational structures (and their associated programs) will fare if we are in a long era of declining federal funding and internal cost cutting.