I've heard of many satellites talk about a "Safe Mode". What is safe mode, and what are some common methods to entering safe mode?
As mentioned by geoffc, it all depends on the manufacturer and especially the mission.
In the case of the geostationary spacecrafts, the vehicle will usually switch the safe mode only in a contingency situation and after not receiving any ground signal for a given timeout (often 24 hours). In this case, the spacecraft will turn on the omni-directional antenna, start looking for the sun in order to both charge its batteries via the solar panels* and attempt to determine the position of Earth based on its current date time knowledge and ephemeris.
Once Earth is found, the vehicle will stay locked on Earth (with either Earth sensors or using only Star Trackers, like on the latest iteration of the Orbital Sciences Corp. Star 2 platform). The vehicle will usually require manual intervention before resuming normal operation.
Concerning geostationary spacecrafts, it's important to note that a software glitch will not put the vehicle in safe mode. Instead it will switch-over to another flight computer and continue operations, often times without service disruption (although there will be a telemetry and telecommand disruption, this will not affect transponder functionality). In fact, spacecraft are designed in such a way that each subsystem can be relatively independent (as long as the vehicle is powered of course). Each subsystem is connected to a Remote Unit Interface, and that RUI is connected to the on-board computer which will perform TM/TC processing, baseband commands, etc. In other words, the on-board computer (which hosts the flight software) is mainly used to relay telecommands to the appropriate Remote Unit Interface and to aggregate RUI telemetry, generate the correct bit sequence, encrypt the frames and send the frames as such to the TTX in order for the appropriate signal to be generated, amplified, routed to the output multiplexer and emitted through the appropriate antenna.
I hope this answers your question.
(*) The spacecraft will charge the batteries no matter the current state of the charge, recent vehicles equipped with Lithium Ion batteries only rarely require trickle charge, which is not the case for previous generations.
Sources: my previous job (Operations Architecture, SES), my passion and my very much loved copy of The New SMAD by Wertz and co.
With no one around to hit the power switch on site, they need a way to reboot, reset, recover, from any potential error. A Blue Screen of Death in space, can really mean death.
Safe mode, one assumes is a pretty low level hardware thing (I am sure it is implemented different by every manufacturer), where when all else fails, or certain errors occurs, the system can get itself into a state, that the ground can tell it to recover from.
Often this involves pointing the antenna back at Earth, or listening for a specific frequency for a reboot/restart/reset command. It is usually a pretty simple, but specific command (would not want to trigger it by accident). Simple so the remote device can pick it up with minimal functionality, and specific to make it hard to happen when not desired.
It may involve a command to restart on the backup hardware (B side?) or it may just trigger a reboot of the software.
The level of error handling we can tolerate on our devices is no where close to the level you can tolerate on a 100 million dollar device, where no one can reach it directly.