There are over 120 champions and each of them are unique. The importance of knowing your enemy's champions and your champions abilities, and how to use that information to your advantage?
Knowing your opponents, teammates and your abilities is half the battle, for beginners this may take a long time to know all the champions abilities off hand. You may know some champions off hand but it is important to know all of the champions abilities to minimize the chances of you losing. All champions have four abilities and a passive, if you are able to memorize all of the champions abilities and passives, you will have a better understanding of how to fight them or play with them.
The things you should try to memorize about the champions in the game are, what there 3 basic abilities can do, what there "ult" can do, have an idea of how long there "cd" (cool downs) are and an idea of how much damage the abilities can do. Knowing this four things about a champions will help you play against them and also will help you play with them. To find out all of this things "riot games" the company that made the game, has made a spotlight for all champions showing you what the champion abilities are and where to put it skill points into an ability and recommending how to build them. These video's are very helpful for learning a champion, and can be found on Youtube just type the name of the champion then spotlight, then watch the video made by league of legends.
Knowing what champions are capable of doing can help you "trade" better in the laning phase of the game, and if you know how long there "cd" are it may make it much easier to out "trade" them. Knowing all the champions abilities/passives is only half the battle, the way you use this information is what will benefit you and may lead to you winning the game. Knowing that you can out "trade" your opponent will allow you to play more aggressive, this will make it more difficult for your opponent to farm, but if you know your opponent can out "trade" you then you may want to play "passively".
Taking the time of learning all the champions will be time consuming but will be a lot easier than trial and error, so take the time to read over their abilities/passives. To find where to read/watch all the champions abilities/passive open up league of legends, then click view/edit summoners profile at the top right, then click champions, then pick/click a champion you want to learn about, then go to abilities and you should be able to watch the champion abilities/passive. You can also get tips on how to play the champion or tips on how play against that champion. Just click tips on the pages and it will pull up the page then you have to click a champion.
Note: Do not try to learn all the champions abilities in one day, it will overwhelm you and may confuse you, just take the time to learn one or more champion's) every time you play. This will build up your memory of knowing the champions in the game and will help you win games. Some champions may change over time when an update comes out, read the patch notes it will help you understand the changes that have been made.