DPS has been a common term in the gaming community for as long as we can remember. It is a role in RPG games where a character or a champion serves as the main source of damage to his/her entire party. DPS champions usually pack heavy damage be it physical (with swords and bows) or magical, and the same can be said about it in League of Legends.
DPS champions are the backbone of every team fight because they are the ones that clean up the battlefield, they are mostly comprised of Bruisers, Marksmen, and Slayer champions.
DPS champions are commonly ADC or bot lane roles as well as mid lane and occasionally top lane, these champions provide Damage Per Second, hence the acronym DPS. In League of Legends, not all champions that deal damage are considered DPS unlike in common MMORPG games, DPS in League of Legends are champions that can deal high damage each second, we emphasize this because there are also damage champions that can deal burst damage.
Champions like Assassins or mages who can take enemies out in one second. DPS champions, on the other hand, brawl and play with their opponents and kill them in strategic ways, most DPS champions can take out multiple enemies at once unlike assassins that can only deal with one enemy at a time.
From the word itself, Damage Per Second, it is the damage your champion can do in one second and it can mostly be improved by increasing your attack damage and your attack speed.
Take Jinx for an example, she is a DPS champion at the ADC role or a marksman, she has an attack speed of 1.00 and attack damage of 150, which means she can deal 150 physical damage with her basic attack every second, now if she increases her attack speed, like for when she kills an enemy and activate her Get Excited! Passive, she will increase her attack speed therefore she can make more attacks within a second, hence becoming an even stronger DPS champion. In simple words, if you have a 2.00 attack speed, you will be dealing 300 Damage Per Second.