1) If your trigger is running in before update then you only need to fill the value in field (if field is on same object) 2) If you are updating any other object's field then as Avi suggested. 3) Make sure you don't write update explicitly in after update as it may give you exception of "record is read only"
Better if you can post your code here, and let us know what exactly you want then we may help better.
http://www.sfdc99.com/2013/05/12/example-how-to-write-a-simple-apex-trigger-2/
SO may be you just need to do a DML ?
2) If you are updating any other object's field then as Avi suggested.
3) Make sure you don't write update explicitly in after update as it may give you exception of "record is read only"
Better if you can post your code here, and let us know what exactly you want then we may help better.