Do changes in an organism's cell modify the genetic information it uses for reproduction?
In multicellular organisms there are specialised cells called gametes that are responsible for sexual reproduction (sperm/ egg cells). Mutations occurring in these cells can be passed on to offspring (germ line mutations). Mutations occurring in others cells affect only the given individual (somatic mutations).

Edit:
For unicellular organisms it is pretty straight forward, the daughter cells formed through fission are identical. As for multicellular organisms here are many forms of asexual reproduction, and even these may involve the formation of specialized cells (eg. spores). So my answer is: it depends on the type of asexual reproduction and the actual cell(s) that accumulate mutation(s). For example in parthenogenesis - where the unfertilized egg cell develops into a new individual - , a mutation can be inherited if it occurs in the germ line cells generating the egg cell, but any somatic mutation would stay in the parent. You can find detailed info on asexual reproduction with details on each form on this wiki page