As imparted by joelgundam01 in the "What manga are you reading/just finished reading?" topic in the "General Manga & Light Novel Discussion" forum:
Onii-san = Big Brother
Onee-san = Big Sister
Actually that's not 100% accurate. In the case of the post I made in the manga thread, my comment was indeed 100% true. However, that's not always the case.
Technically "Nii" is Big Brother and "Nee" is Big Sister. San of course is just a standard honorific, but "chan" and "kun" could also be used.
Onii and Onee are used when the younger sibling has high respect to their older brother/sister or the older sibling has a high status/reputation in society. For example, a noble or royal will call their older siblings "Onii" and "Onee" along with the "sama" honorific at the end.