Roughly, a belief that a scientific
description is the most appropriate description
for any situation.
The way I use this term (which might not in fact be the way it is most commonly used in philosophy—but it's a highly loaded term anyway, only used of one's opponents) does not imply that the belief is unscientific, nor that it has an unsupportable basis in faith, nor even that the description is untrue.
A "scientistic" description might consist of a true description of the physical facts. However, I would label it as scientism if I thought it was an inadequate or inappropriate description.
Of course, if you believe that there exists something other than what is scientifically describable (e.g. mind, God, or moral absolutes), then necessarily a scientific description is inadequate. However, the mainstream philosophical view is materialism, without any of these extras, and in this case it is not that the analysis given solely by science is incomplete, rather that it is sometimes not helpful.
For an example, take something like a street sign, or a scrubbing brush, or a floral bedspread. A scrubbing brush is made of atoms—we could get picky about this, but let's agree it's nothing but atoms—of plastics, hydrocarbons, and its shape is essentially a cylinder distorted in such-and-such a way with such-and-such geometric shapes stuck on. A bad way of reading "Here is my scrubbing brush" is claiming that this is shorthand for the more accurate "Here is my distorted cylinder of hydrocarbon molecules with...".
If I'm looking at a red book, what I'm actually doing is looking at a red book. But a bad description of it is saying that I'm seeing a red "patch", or that light rays of such-and-such a wavelength are entering my eyes and causing such-and-such potentials in my nerves and brain. All this might be true; and there might be nothing else going on in me apart from these physical reverberations—but it is looking at the situation in the wrong way. It is not a more accurate form of description.
Yes, all our objects are made of atoms, and all our thoughts are electrochemical changes in our brain, almost everyone agrees. The error is in underestimating ordinary language: in supposing that the expressions "seeing a red book" or "scrubbing brush" are pieces of crude folk metaphysics which need to be clarified and pinned down to a skeleton of physics and chemistry.
In this sense in which I use it, philosophers who indulge in scientism include W.V.O. Quine, Donald Davidson, and Patricia and Paul Churchland. They tend to derive from (or in Quine's case once belonged to) the school of logical positivism, which was based on the atomism of Wittgenstein's early book Tractatus Logico-Philosophicus.
Those who correctly oppose scientism include Gilbert Ryle and Daniel Dennett. Their tradition is the more linguistic one of analytic philosophy, of which Wittgenstein's later Philosophical Investigations is the central work. But Dennett is very scientific and materialist, and both he and Ryle are strongly eliminativist in that they reject a lot of non-physical things.
Again, a caution that this usage might not be that encountered in other philosophical writings; and any labelling with the term will be contentious.