From the Star Wars movies it appears that whether a lightsaber "ignites" is dependent on the circumstance but is there an explanation for how they turn on (i.e. a simple switch on the side or through the Force)?
It varies, and is completely up to the character who designs/builds the lightsaber.
The majority of lightsabers have some sort of external switch for activation, with said switch being a "dead-man's switch". As long as the switch is depressed, the blade remains activated. The switch itself - called an activation stud - could be anything from a lever or button to a plate.
Of course, if needed this switch can also be activated using the Force... in which case, the lightsaber would activate but then immediately de-activate after the switch is toggled. This technique was famously used by Mara Jade to kill Dequc, the brief leader of Black Nebula. Since she could not physically reach Dequc, she instead hid her lightsaber inside a statue and activated it from across the room while Dequc examined it:
However, there have also been examples of designs in Legends canon where the switch was internal, thereby requiring the Force to activate. These designs were rare, since although an internal switch is more secure, it requires constant focus in order to maintain the blade activation.
In addition, at least one lightsaber - that of Lord Hethrir - was built with NO activation switch at all, instead requiring the wielder to complete the circuit themselves through the Force:
The whining hum of Lord Hethrir's lightsaber filled the room, and the silver-gray light of the blade cast shadows on Tigris's empty hands. Tigris raised his head, to gaze in wonder as he always did at the radiance of Lord Hethrir's saber.
The blade vanished.
"Try once more," Lord Hethrir said, and gave the handle of the lightsaber to Tigris. The handle of the saber felt warm in Tigris's grip. The lightsaber was too large for Tigris's hands, but he clasped it as best he could.
He knew what Lord Hethrir wanted him to do.
The blade of Lord Hethrir's lightsaber could only be activated by the use of the Force. Hethrir would not accept anyone into his inner circles who could not complete the circuit and generate the blade.
Tigris tried, how he tried, to make a connection to the Force, to extend himself, to create the blade.
It is activated by a switch.
Directly from the script for A New Hope
Ben hands Luke the saber.
LUKE What is it?
BEN Your fathers lightsaber. This is the weapon of a Jedi Knight. Not as clumsy or as random as a blaster.
Luke pushes a button on the handle. A long beam shoots out about four feet and flickers there. The light plays across the ceiling.
It can be noted that this is immediately before Luke has even ever heard of The Force, so it is highly unlikely that The Force is required in any way to activate the lightsaber. Additionally, General Grievous is able to wield four lightsabers simultaneously, and is not at all Force-Sensitive.
From the novelization of The Empire Strikes Back
The Wampa Ice Creature spread its black, hooked claws and lumbered toward the hanging youth . Suddenly the lightsaber, as if by magic, sprang to Luke’s hand. Instantly, he depressed a colored button on the weapon, releasing a bladelike beam that quickly severed his icy bonds.