Rifles come in many shapes and varieties. Essentially, a rifle is a firearm with a rifled barrel that has a butt stock that must be placed in the shoulder to shoot. While there are always exceptional examples, we'll stick with the more common actions here. The "action" is as accurate a description as it gets - that's the part of the rifle where all the action happens. It is a meta term for lots of other parts of the gun, including the receiver and barrel (which includes the chamber).
There are lots of ways to categorize rifles. Some are powered by air and shoot pellets. Some are sized to shoot a round that can travel over a mile. Some load the next round for you. Some require that the shooter open the action to eject a round and load a new one. There are cowboy specials, hunting guns, and Olympic style competitive paper punching equipment.
The bolt action is one of the more popular rifle choices for sport shooting. The bolt is a sort-of-cylindrical piece of metal (usually steel) that rides in the receiver. It has a handle that is closed to shoot. The front of the bolt, called the cartridge seat, butts up against the rim of the cartridge to contain the cartridge upon ignition. The firing pin assembly usually resides in the bolt and is released when the trigger is pulled. After firing, the bolt is opened and the extractor pulls the case out of the chamber. When the bolt is pulled back, the ejector pops the round out of the receiver. The next round is ready to be loaded, either from a magazine or in the case of a single-shot rifle, by the shooter.
I shoot bolt action rifles primarily. The Anschutz target rifle (19-series and 20-series), Chris Kyle's favorite rifles, and the American Rifleman's favorite hunting rifles which include the Remington 700 and Winchester Model 70.
The semi-automatic rifle is built on the same idea as the semi-automatic pistol - using the physics principal of "equal and opposite force" to operate the bolt and load the next round. There are a variety of mechanisms to accomplish the bolt operation. Some are direct blowback, where the gas that propels the bullet forward is used to push the bolt backward. Some are gas operated, where gas is contained in a closed system to push the bolt backward.
This is a classic "cowboy" rifle. Bullets are typically loaded in a tubular magazine that is parallel to the rifle barrel, and each bullet is loaded by the user pulling on a lever. That lever is typically underneath the action and the motion to pull the lever is "downward" from the action. Pulling the lever opens the action and ejects the used case from the chamber. Closing the lever again will load the next round into the chamber. The lever action rifle has timeless appeal. Lever actions are still used frequently for target shooting and hunting.