The exact approach may vary from engine to engine but the overall theory is basically as DrSuperSocks has suggested. You basically hold an array of data that would explain what tiles are where in the level map, most of the time you use something of a camera class to track where you are in the level and using those measurements you render the tiles to the screen appropriately. Depending on the complexity of the game sometimes additional arrays or more complex map objects are used to also store information for event triggers and collision responses and such, however this is entirely up to you as the designer / programmer on how to best implement what you would like to accomplish.
The good news is that the side scroller / platformer type game is the generic "hello world" tutorial of most engines and development kits. I would suggest that you do some research on what engines are available for the platform(s) you wish to target and see what engine you would like to use first. From there you can then start working your way through the engines tutorials to get familiar with the engine and it's practices and almost inevitably learn how to create a platformer or side scroller with said engine.
On the same note if you are considering writing your own engine to power your first game project I would have to highly recommend against it. I hope it doesn't come off as rude but if you are confused on how to implement tiling and or platforms, chances are you are not quite ready to undergo the complex activity that is creating a gaming engine. As such it would be best to first do your project using an established engine and or development kit and get a feel for how the engine you choose accomplishes the goals you are looking for. After completing your project it will be much easier to go back and create your own custom engine that emulates the practices you have used previously.