, a factive expression (word or phrase) is one that asserts that something is a fact
: in other words the speaker is claiming that what the expression governs is true.
The usual use of 'factive' is with verbs like 'know', as opposed to non-factive 'believe'. If John knows that Mary has bought marshmallows, then it entails that Mary in fact has bought marshmallows. If John believes that she has, we don't know whether she has or not. No matter how sure John is, and whether or not he says 'I know she has bought them', if she hasn't bought them, then he's wrong: he doesn't know it, he merely believes it strongly.
This is not a philosophical point about the fallibility of purported knowledge, but a grammatical point about what the use of certain words entails. If I state that 'Mary knows that John is a drunkard' then I am equally committed to the assertion 'John is a drunkard'. I'm not licensed to use the word 'know' when I'm merely asserting someone else's belief or certainty: I have to believe and be certain of it myself, even though I'm only reporting someone else's thoughts.
In English there are other factive verbs: they include 'realize', 'regret', and 'discover'. If you regret that it's raining, then it must be raining. So, once again, it's not some peculiar metaphysical property of knowledge being invoked. It's baggage carried by the word. Some linguists have analysed it as carrying an extra element at the head of the subordinate clause.
Apart from verbs, other constructions may be factive or non-factive. The complementizer 'the fact that' is itself factive: if I state 'The fact that you are late does not surprise me' then I am committing myself to 'You are late' (which I am not in 'I expected you to be late' or 'I presume you are late'). The adjectives 'alleged' and 'putative' make a qualification explicitly non-factive.
The qualification is always of some embedded component of the sentence: a noun phrase like 'alleged burglar', or subordinate clause like 'that you would be late'.
The whole sentence is always in a sense factive, though the term is not used of whole sentences. By a pragmatic principle, one of Grice's maxims, it is assumed you are telling the truth when you say something: that is, you believe what you say; you think it is a fact. This applies to the indicative or declarative mood, but not to other moods like questions or commands or the subjunctive.
Many languages often have something like a subjunctive mood to convey that what it says is entertained, supposed, possible, wished-for, or even false and past hoping for. This contrasts with the indicative mood of simple fact-stating. In English the verb 'hear' is not factive, and our subjunctive is only a remnant: we say 'John heard that Mary bought marshmallows' and it's not clear whether she did. In Icelandic this is factive if 'bought' is in the indicative, and non-factive if it's in the subjunctive, a rather neat division, I think. See also French subjunctive for a similar distinction.