This depends a little on the device and what version of Android you have.
On versions of Android later than 4.1 you can access safe mode by bringing up the power down menu (i.e. hold the power button) and long pressing the "Power off" option. This will bring up a little dialog box that will ask if you want to reboot into safe mode. The addition is noted on the Android web site under "System" changes.
On some older phones - though not all will necessarily support this - there may be a button combination that you can hold at boot to get into safe mode instead. For example, the Galaxy S2 used to allow you to hold down the hard menu button during the boot process to start up in safe mode. Others have used (I believe) holding both volume up and volume down. Check your user manual (or Google) for your specific device to find more information, including whether or not the device supports safe mode at all.
As to what "safe mode" means - it generally disables third party apps and widgets, to help you recover from a bad app install, give you a chance to remove a possibly rogue/malicious app, or simply determine what apps may be causing problems in your system.