The most common way of splitpushing is with 1 person on a sidelane and the rest of the team grouped up far away, tying down the rest of the enemy team. In order to do this, your team has to fullfil the following criteria:
-your splitpusher is stronger 1v1 than anyone else on the enemy team and is able to deal damage to the enemy tower even with an opponent in his lane OR
-your splitpusher has a teleport up, a reliable way to teleport away without taking too much damage, the enemy solo defender doesn't have teleport AND you have large enough advantage to get an objective in the short time you are 5v4.
I need a hero!
In order to splitpush, you need to have the right solo laner. Requirements are powerful 1v1 lategame and the ability to escape or at least survive a very long time if ganked. The 2 best splitpushers currently are Jax and Tryndamere, both posses short cooldown jumps and both win 1v1 if equal on gold against almost anyone.
Nothing as sad as a death unforseen
Vision of the enemy jungle is essential when splitpushing: splitting up can make getting caught a lot more punishing. If you have trouble getting vision deep in the enemy jungle, try grouping up as 5 first, warding and only then splitting up. Keep your eyes on the minimap, if an enemy has disappeared, play careful until you have a decent idea whore they are. Even if the damage you deal to towers is just a little bit every wave, keep at it; it will add up quickly.
So those are the very basics of splitpushing, happy splitting and merry pushing!