Ontology is the study of what there is, an inventory of what exists. An ontological commitment is a commitment to an existence claim.

Ontology vs. metaphysics
Although the term terms "ontology" and "metaphysics" are far from being univocal and determinate in philosophical jargon, an important distinction seems often enough to be marked by them. What we may call ontology is the attempt to say what entities exist. Metaphysics, by contrast, is the attempt to say, of those entities, what they are. In effect, one’s ontology is one’s list of entities, while one’s metaphysics is an explanatory theory about the nature of those entities.
For example, a philosopher might include in her ontology quarks, people, substances, and institutions, while her metaphysics might include such claims as: quarks are necessarily such that they obey thus and such physical laws; people are essential rational beings; substances are by nature indestructible; institutions are essentially conventional entities; and so on.
The received view of ontological commitment
A theory of ontological commitment is a theory that tells us when we are committed to the existence of certain entities. Why do we want a theory of ontological commitment? Applying such a theory gives us a way to move from sentences commonly accepted as true to more contentious claims regarding what there is in the world. If we can show how to move from a list of true claims to a list of what exists, we can anchor our ontological claims on firm ground.
The most familiar theory of ontological commitment is that offered by Quine in his "On what there is" (1948). It may fairly be called the received view of ontological commitment. In effect, it is a combination of a criterion of ontological commitment and an account of that to which the criterion applies.
The criterion itself is quite simple. A sentence S is committed to the existence of an entity just in case either (i) there is a name for that entity in the sentence or (ii) the sentence contains, or implies, an existential generalization where that entity is needed to be the value of the bound variable. In other words, one is committed to an entity if one refers to it directly or implies that there is some individual which is that entity.
Quine’s account of that to which the criterion applies provides the theory some bite. On his account, a sentence is not, in fact, committed to an entity if there is some acceptable paraphrase of it which avoids commitment to it as per the criterion.
The appeal to paraphrase allows us to avoid the problem of Plato’s Beard, or the problem of nonexistent entities to which we nonetheless apparently refer. The names are to be eliminated in such a way that the remaining set of true claims contains none committed to any such entity after the manner of the theory. For example, the name ‘Pegasus’ is eliminated in favor of a verb ‘Pegasize,’ which is understood as the thing one does when one is Pegasus. We can then say that nothing Pegasizes.
The received view also makes it easier to deny the existence of universals. Merely using predicates does not commit one to universals or properties, as something of the form "a is F" neither requires one to name a property being F or quantifies over any such thing. This may be counted as an advantage, in that it is not as if any piece of language used must have an entity corresponding to it. If one is committed to universals, it must be by virtue of subscribing to sentences like "Courage is a virtue," which do refer outright to such things.
Ontology and psychology
In the philosophy of mind, theories of ontological commitment come into play when considering true psychological claims. Consider the following psychological claims:

       (1) Jones saw a red image.
       (2) Smith believes that the sky is blue.
Each of these seems to imply a distinctive ontological commitment given the received theory’s criterion. A natural way to render them canonically is as follows:
       (1a) There is some x such that: Saw(Jones, x) and x is a red image.
       (2a) Believes(Smith, That the sky is blue)
The first quantifies over red images and the second uses a name for something akin to a proposition. Both commitments may be found problematic.
A claim like (1) can be true even when Jones is hallucinating and there are no red physical things in the situation. If we accept that (1) commits us to red images, we are then stuck acknowledging a red nonphysical thing, making materialism apparently false.
One standard attempt to avoid this commitment is adverbialism (See Tye 1989). On this approach, instead of construing (1) as relating pairs of entities, it is construed as predicating, of a single entity, a complex attribute. What looks like a second entity being related to the first is rather a modification of the predicate, that is, an adverb. Instead of (1), then, we have something like:
       (1b) Jones saw redly.
Of course, the English mangling of (1b) is neither here nor there. The point is that the form of (1) could be understood as parallel to that of:
       (3) Jones ran slowly.
In the case of (3) a predicate modifies another predicate and only one entity is implied, namely, Jones.
The payoff of adverbialism can be found in the hope that it will be easier for the materialist to accommodate a world in which people sense in certain ways than he could a world in which such things as mental red images exist. If (1) is understood as (1b), it does not imply the existence of any red entities.
The adverbialist project requires more than a willingness to mouth such odd bits of mangled English as "Smith believes in a sky-is-blue kind of way." It also requires that one show how such paraphrases can do all the work the original sentences did, and this may be a tall order indeed. It is not clear that they can capture the same distinctions and implications the originals do. See Jackson (1977) for a powerful critique of the approach applied to phenomenal objects.
Propositional attitude attributions like (2) seem committed to something like propositions, and this, too, may be undesirable. Propositions may be thought problematic due to their general abstract character; or, more significantly, they may be thought problematic due to the difficulty of arriving at a suitable theory of what a proposition is such that it could be the thing standing in those relations. If propositions are construed as sets of possible worlds, for instance, believing that 2+2=4 ends up being the same as believing the Pythagorean theorem, an inadmissible result. Other accounts coordinating theories of propositions with objects of belief encounter their own worries (See Schiffer 1987, chapter 3). While the adverbial strategy has been most famously applied to the case of phenomenal object, it can be, and has been (by Tye 1989), applied to propositional attitudes.
Troubles with the received view
The received view of ontological commitments faces at least three important worries concerning the status of paraphrase, its adequacy in capturing ontological concerns, and the inscrutability of reference.
The status of paraphrase. The first difficulty turns on the standards governing adequate paraphrase. One problem is that it is unclear how far paraphrase can go in eliminating ontological commitments. Could we not, perhaps, construe all sentences on the model of "it’s raining," so that nothing is referred to at all? Perhaps "There is a cat on the couch" should be paraphrased as "It’s cat-on-couching," or the like. Given the contortions of adverbialism, one is entitled to wonder how far, and how arbitrarily, commitments may be avoided through paraphrase (see Ackerman 1995). A related but distinct worry is that the theory seems to face a fatal dilemma: either the paraphrase is equivalent to the original, or it is a replacement thereof. If it is equivalent, then it is unclear how the paraphrase, as compared to the original, can be privileged with regards to its ontological commitments. If the original was committed to Pegasus, and the paraphrase is equivalent, why should we not conclude that the paraphrase is also committed to Pegasus? On the other horn, if the paraphrase is not an equivalent but a replacement of the original, then the truth of the original is denied, and the advantages of a theory of ontological commitment are lost. As said above, the idea is to find those claims that are on all hands agreed to be true and then discern their ontological commitments (Jackson 1980). In response to this worry, Jackson has proposed a modification that in effect replaces paraphrases with metalinguistic statements about the referential apparatus involved in the original sentences. On that modification, no equivalence is claimed, nor is the original sentence rejected as false; rather, the metalinguistic statement is "privileged" to give the commitments in the sense of providing an account of the semantics -- and hence referential implications -- of the sentence.
Mundane vs. categorical ontology. The second worry is that this theory wrongly classifies ontological disputes in philosophy as on a par with more mundane disputes. Norton (1977: 89) has asked us to compare such questions as:
       Is there a pencil on the desk?
       Is Jones having a blue after-image?
       Is there a prime number larger than one million?
       Is there a class made up of all the things in this room?
with such questions as:
       Are there any material objects?
       Are there any sense data?
       Are there any numbers?
       Are there any abstract entities?
On the face of it, there is some principled difference between these kinds of questions. The latter are typically philosophical, while the former are not. On the received theory, the questions are all of a single kind, which implication may be counted a defect.
Carnap (1950) presents a theory of ontological commitment that distinguishes these groups. What he calls internal questions correspond to the first group; external questions correspond to the second. Internal questions are asked within a given linguistic framework; the decision has been made to use certain linguistic rules and the question is to be decided on empirical grounds. External questions concern the adoption of linguistic frameworks. Certain linguistic rules make it analytic that there are, for instance, material objects. The choice between different frameworks is made on pragmatic grounds only; there is no sense to saying that a choice of linguistic framework is true or false, only that it is more or less useful. On this view, then, there are two kinds of ontological commitment: the internal kind, which Quine’s view seems intended to capture, and the external, which in an important sense is conventional. Of course, Carnap’s theory appeals to the notion of analyticity; those who reject the analytic/synthetic distinction must reject Carnap’s theory.
The inscrutability of reference. Finally, the third problem facing the Quinean theory comes from Quine himself. The doctrine of ontological inscrutability is the doctrine that, given a class of truth-valued sentences, there will be more than one assignment of referents that will produce the right truth-values such that there is nothing to choose between them. If this is correct, however, then there is no fact of the matter about what the ontological commitments of a set of sentences are; the various assignments of referents provide a disjunction of such commitments, none of which can be said to provide the true commitments of the theory. Quine addresses this worry in his "Ontological relativity" (1968), conceding that in some sense there is no fact of the matter about the ontological commitments of a theory.
Gene Witmer