During suspend most devices are shutdown, and system state is saved in RAM. The system still requires power in this state. Most modern systems require 3 to 5 seconds to enter and leave suspend, and most laptops can stay in suspend mode for 1 to 3 days before exhausting their battery.
During hibernate the system is fully powered off, and system state is saved to disk. The system does not require power, and can stay in hibernate mode indefinitely. Most modern systems require 15 to 45 seconds to enter and leave hibernate, and entering and leaving hibernate takes longer when you have more memory.
Hybrid-suspend is the process where the system does everything it needs to hibernate, but suspends instead of shutting down. This means that your computer can wake up quicker than for normal hibernation if you do not run out of power, and you can resume even if you run out of power. s2both(8) is an hybrid-suspend implementation.