A variety of things in short though it is a very safe guess to say that the majority of the victims were cremated.

The extermination camp Auschwitz-Birkenau as well many other "regular" concentration camps had crematoria (such as the Auschwitz Stammlager, Majdanek, Mauthausen etc.). In Auschwitz Birkenau for example the gas chambers and crematoria formed one complex (espeically with Crematoria III and IV where even the Sonderkommando - the prisoners in charge of burning the body and cleaning the chamber practically lived in the complex itself). All in all Auschwitz had 52 crematoria ovens that could burn up to 6.000 corpses every 24 hours. This idea stemmed from the T4 Euthanasia facilities where handicapped and mentally ill people had been murdered between 39 and 41 and cremated afterwards.

In the Operation Reinhard camps though the victims were intially buried in mass graves, similar to the Einsatzgruppen mass shootings in the Soviet Union and elsewhere. This presented a problem, especially in the extermination camps since as is described by several people who were perpetrators there during the summer the head lead to the remaining bodily fluids rising above earth surface and emitting a terrible stench. Partly due to this and partly because the Nazis wished to cover their tracks from March 1942 onward, the so-called Sonderkommando 1005 travelled from mass execution site to mass execution site to dig up the bodies and burn them over pits of burning material. The ash was subsequently reburied in the soil and in some places like Sobibor archeologists have been digging up what is essentially the ashen remains of hundreds of thousands of people.

Others were simply buried or left where they had been killed such as in the cases of death marches. Sometimes the Allies were able to locate them and give them a proper burial and sometimes not.

As for the ashen remains, as I said above, sometimes they were reburied and sometimes, such as was the case at the Hartheim euthanasia facility, the ash was dumped in the nearest river.

Sources:

• Arad, Yitzhak, Belzec, Sobibor, Treblinka, Indiana University Press, 1992.

• Shmuel Spector, Aktion 1005 — effacing the murder of millions Oxford Journals, Holocaust and Genocide Studies. Volume 5, Issue 2. pp. 157–173.

• Piper, Franciszek (1998) [1994]. "Gas chambers and Crematoria". In Yisrael Gutman; Michael Berenbaum. Anatomy of the Auschwitz Death Camp. Bloomington, In: Indiana University Press. pp. 157–182.

Edit: Here you will find the excavation report from Sobibor in 2013 by Yoram Haimi. It includes their findings as well as photos (heavily NSFW)