We leverage a number of different systems for connecting people in the field. As Frank pointed out the initial first wave of connectivity that arrives is usually BGANs. Reason for this is that they can be hand carried in and set up pretty easily without requirement for licensing, etc. (in most countries). The main problem with BGANs is that you pay around $6+ per Mb of usage.
The next wave of satellite based connectivity we leverage are VSATs. These come different shapes and forms, but usually take some time to install and set up. They also require licensing from the host government in order to be used. The cost of setting up and installing is still relatively high, but the good thing is you pay a fixed price for the bandwidth, regardless of use (in the thousands of dollars though in most places).
The future is looking towards making all of the above easier, cheaper, smaller. But there are also interesting R&D efforts under way such as Google's Project Loon (using weather balloons to deliver connectivity to rural areas) and Facebook's Drone based internet (similar concept to Loon - just using high altitude drones).
If you look under my publications then you can see description of how these things were used in the 2010 floods in Pakistan.
Nowadays, SatCom On-The-Move (SOTM) is coming into operation. Applications for SatCom On-The-Move (SOTM) are intended to provide mobile users with communication services especially at places without any terrestrial communication infrastructure, or at disaster scenarios.
Conference Paper On-Earth Performance Evaluation of SatCom On-the-Move (SOTM) Terminals