Presentism is simply a view in the philosophy of time -- the view that only the present exists.
Presentism has been encountered of late in discussions of Open Theism, where it is said that "God's knows all things that are knowable, but not things that are impossible to know. It is impossible to know the future choices of being with free-will (which God has freely chosen to create), therefore His knowledge includes all past events and that of the present, but none in the future save what He plans to do." 
While many open theists hold to presentism as defined above, many presentists do not deny God the exhaustive foreknowledge that this quote denies. Many philosophical presentists believe there to be truths about the nonexistent future. All presentists believe there to be truths about the nonexistent past, so it makes little sense to say that there are no truths about the future merely on the grounds that it doesn't exist.
As a matter of sociological fact, very few philosophers who deal with the metaphysics of time consider presentism a viable view given what we know about spacetime in physics, though a few of its defenders are trying to salvage it in the face of such objections. It's helpful to think of presentism in contrast with some other views about time. Eternalism is the most popular contender, and the primary debate is between those two theories. A few philosophers have defended other views in addition.
Presentism and eternalism differ about the existence of times. Both views agree that the present exists. Eternalists also believe that all past and future times exist. The whole timeline exists, and everything at every moment has just as much existence as what's existing now. Presentists believe that only things that exist now exist at all. There's no possibility, according to presentism, of something's existing in the past or existing in the future. They will say that something did exist in the past or will exist in the future, but nothing in the past exists in the past. It exists now, or it simply fails to exist.
One alternative sees time as a growing block. Once a moment in time exists, it remains in the realm of being. New moments come into being, and time grows as new moments add on to the end of the timeline. The primary motivation for a growing block view is to capture the sense that the future and the past are not analogous. In fact, one motivation people give for presentism is actually not really a motivation for presentism but (if the argument is any good, and that's controversial) a motivation for a growing block view. That motivation has to do with future contingents (see Open theism for more on future contingents). If statements about the future are either necessary and true or simply neither true nor false, then no contingent truth (a truth that didn't have to be true) can exist with regard to the future. If the past is not like this, and a metaphysical view of time is supposed to ground this fact, then the growing block view could do that. Presentism could not. The downside of the growing block view is that the past and present have all the features of eternalism involving multiple times existing. Presentists think this view is as bad as eternalism in that way. It also denies the existence of the future, which eternalists will resist on the same grounds that they deny presentism. One might consider it the worst of both worlds, therefore, unless this idea of the past and future not being analogous is so important as to trump all other concerns.
Another alternative might be called a moving spotlight view. The whole of time exists, but there's some special ontological status for the present. If this involves degrees of being, with the present being more real but all of it existing, it denies standard philosophical views about existence. It becomes very difficult to do formal logic if 'exists' doesn't simply mean that something exists, an all-or-nothing matter. If it doesn't involve degrees of being, and the present is only special in some non-ontological manner, then all of time exists, and eternalism is correct. If it doesn't involve degrees of being but is about ontology, then it seems there's something ontologically special about the present, i.e. it exists, but that means the past and future don't, and you have presentism. So the moving spotlight view has problems even in terms of how to express the view in a way that doesn't just reduce to another view.
Most arguments for presentism appeal to common sense. Most arguments against it appeal to science or to semantical considerations about how to speak about the past and future if the things you're talking about no longer exist or don't yet exist. If there's nothing in reality to refer to, then what are you talking about? Presentists have to paraphrase such statements or otherwise explain what makes such statements true, and that task has proved formidable. This problem also arises for the growing block view.
- Richard Chappell, "Now and Forever" (argues that presentism is inconsistent with relativity in physics)
- Ned Markosian, "Time" in the Stanford Encyclopedia of Philosophy (general introduction to this and other issues in the philosophy of time; Markosian is a presentist)
- Ted Sider, "Presentism and Ontological Commitment" (Sider is not a presentist, but he here defends presentism from the charge that presentism cannot consistently speak of the past and future by appealing to a notion he calls quasi-truth)