PIMBY is a response to NIMBY -- Not In My Backyard. While the NIMBY attitude is characterized by an unwillingness to accept any negative externalities involved in large public works or large companies (power companies, pig farms, sewage plants, etc.), PIMBY is the attitude that maybe we should replan these institutions to keep all externalities acceptable to the local community.
While this idea sounds pretty straightforward, the term is used with many shades of meaning by many different groups.
PIMBY is most often translated as either "Please In My Back Yard" or "Power In My Back Yard"; it is generally associated with personal solar power panels or wind turbines, where the homeowner produces their own power. It may also be used by those who raise their own food crops, food animals, use composting toilets, use greywater systems, passive solar heating, etc. The idea is basically that instead of relying on public utilities or corporations, the homeowner provides for themselves (at least in part), and suffer any negative externalities themselves. It may also be applied to farmers that set up wind farms on their land to help with their profit margins, or other projects of this sort. This is often strongly associated with the sustainability and DIY movements, although there is no inherent connection.
In the business world, PIMBY often refers to companies that work to reduce their effects on their neighbors to the point that there is no reason to object to them being 'in your backyard', which usually means not only low-noise and low-pollution, but also other Earth friendly habits that are not necessarily apparent to the neighbors -- things like low carbon emissions, energy efficiency, and recycling programs. In contrast to these idealistic PIMBYs, companies may also offer economic or infrastructure benefits to the community to turn NIMBY attitudes into PIMBY -- and in many places, simply the aspect of more employment opportunities encourages PIMBYism.
To confuse matters further, PIMBY may be used in a purely negative sense; you might say that you are not a PIMBY when it comes to fracking, for example.
Despite these many meanings, the term PIMBY has entered into the public memosphere to the point that other organizations have been co-opting it; for example, a Google search finds references to PIMBY pages which turn out to be groups such as "“Pilgrimage In My Back Yard" (a Catholic event teaching the history of local Catholicism) and "“Pug In My Back Yard" (an architectural course for elementary school students, of all things).
The term YIMBY ("Yes In My Backyard") is also used.