In grammar, a small clause is an anomalous construction (though common in English) that bears some logical resemblance to a clause but in other respects does not behave like one. The term is mainly used in Chomsky's approaches to syntax, because other theories analyse them differently, not as any kind of clause.

They occur when the object of a main clause has some kind of complement:

Mary considers John a bore.
Mary finds John attractive.
Mary prefers John out of the room.
Mary sees the bomb explode.
Logically these are equivalent to a main clause plus something like the following finite clauses, used as the direct object of the main verb:
John is a bore.
John is attractive.
John is out of the room.
The bomb explodes.
She doesn't find John, and she doesn't (necessarily) see the bomb. What she finds is that "John is attractive"; what she sees is that "the bomb explodes". They are also equivalent to longer forms with the explicit complementizer "that", which causes the following verb to take on finite tense and agreement:
Mary considers that John is a bore.
Mary finds that John is attractive.
Mary prefers that John is out of the room.
Mary sees that the bomb explodes.
They are also different from cases where the same main verb has another kind of complement that is unequivocally just a complement, not a separate clause:
Mary considers John for the job.
Mary finds John a job.
In a way it seems like the small clause could be created from the longer expression just by omitting material, "that" and the finite verb "is".

Against the clausal interpretation is the fact that it is the main verb that gives case marking to the object. We can see this if we use pronouns:

Mary considers that he is a bore.
Mary considers him a bore.
So the simple omission theory doesn't work. If you just dropped "that" and the inflected form "is", you'd be left with *"Mary considers he a bore". Instead, the verb "consider" clearly governs "him" as an object.

This aspect is evidence for the non-Chomskyan approach, in which we have an ordinary governed object in "Mary considered him", and an adjunct "a bore" or "foolish" or whatever. The adjunct, whichever theory you hold, can be of almost any type: adjective ("foolish", "kind to animals"), noun phrase ("a fool"), verb phrase ("sing", "recite poetry"), or a prepositional phrase ("out of the room").

In Chomskyan theory you have to explain that certain verbs, called ECM verbs (for Exceptional Case Marking) can govern across a clause boundary. The verb "consider" assigns accusative case but can do so to the subject of the clause "he/John [is] attractive", making it "him". Hence the name ECM.

Another problem for the Chomskyan analysis of these constructions as "small clauses" is that they appear to have no verbal inflection of any kind. In normal clauses there is an element I (for "inflection" at the top of the clause, containing either finite or non-finite marking. Finite may appear as the -s of agreement, and non-finite as the infinitive marker "to":

Mary wonders whether John eats carrots.
Mary wonders whether to eat carrots.
If there is no I, then the whole thing can't be an IP, a projection of I. This is a technicality of the Chomskyan approach: the types of higher-level constituents have to be strictly constructed out of types of their components. So if "John foolish" is a constituent with nothing but a noun and an adjective in it, the small clause itself must function as either a noun phrase or an adjective phrase. But the syntax requires any main verb to have some kind of clause around it so that it can assign semantic arguments: in "John [is] attractive" the predicate "attractive" has to be matched with an argument that fills it.

The analysis of these deceptively simple forms is much disputed.

A closely related form is where, as here, there is an ECM verb governing the subject of an infinitive clause, but there is an overt infinitive marker. This is called an exceptional clause:

Mary considers him to be a bore.