Definition from Wiktionary, the free dictionary
See also Appendix:Variations of "han"
Etymology 1 
-han (in words with back vowel harmony; i.e. in words that include a, o, u or i or e with a, o, u—compare -hän)
- (enclitic) A particle to express that the speaker had some (more or less) certain view on something but is surprised after having got to know the truth about it:
- (enclitic) A particle to express that the speaker had some (more or less) certain view on something but is surprised at the controversial view expressed by some other person:
- Olet kyllä turhaan masentunut opettajasi arvostelusta, sillä tuo maalaushan on aika hieno.
- You're sad about the teacher's criticism for no reason, since that painting is quite good.
- (enclitic) A particle to express that the speaker is determined about or resistant towards something:
- (enclitic) A particle appended to an indicative (more rarely: conditional or potential) verb form in order to express the speaker's wish about something; also to politely ask someone to do something—i.e., to avoid the imperative:
- (enclitic) With the particle -ko (ko required by the vowel harmony) to express "I wonder if".
Usage notes 
- As seen above, this particle is used in subjective expressions or opinions and thus especially in spoken language. It can, depending on the context, have several meanings that, in English, can usually be expressed only by accentuating the words differently or by adding extra words.
- Note also that this particle is very often attached to the first or the second word of the clause.
- In everyday speech, this particle is used quite often even if there were no intention to give some special tone.
Etymology 2 
- Alternative form of -hVn. (the case suffix for illative singular case)