Charitable Aid air lift
 

Do charities and NGO's pay for the charter of aircraft to send aid in times of disasters? Or do the air cargo companies donate an aircraft?

I guess that depends.

I know that after the Haiti earthquake and the disaster in Pakistan e.g. Lufthansa Cargo did a couple of flights where the company donated the aircraft. I.e. they paid for the aircraft, crew, fuel, etc.

As a general rule the major aid agencies charter the airplanes they use in humanitarian aid. This is true for agencies like the UN World Food Programme and the International Committee of the Red Cross. Some of the religious charities like Missionary Aviation Fellowship (MAF) or Samaritan's Purse operate their own aricraft.

Some of it is partnerships
 

and some of it is general operations

Plenty of good information with real world examples of TNT's support to World Food Programme here: Moving the World | TNT's partnership with UN World Food Programme (WFP)