League of Legends is a competitive game with a more or less fixed Metagame. This means there are some "Rules" that apply for example the roles in every game (1 Top, 1 Mid, 1 Jungle, 2 Bot). Because of this Metagame things like a tier list can exist.
What is a tier list?
There are actually multiple tier lists. The most popular ones being the ones for soloqueue but even within this category multiple tier lists exist. You also have to know that there is no official tier list. Every list you'll find on the internet is made by theorycrafters, pro players or even amateurs which means that no tier list shows the absolute order to the strength of champions.
There are two main soloqueue tier list categories: FOTM (flavor of the month) and the statistical/opinion based one. The FOTM tier list will look pretty much the same on most websites, since it's just what people play the most while ignoring the actual strength, potential or winrate of the champion.
Trying to adapt to the FOTM tier list usually just serves a psychological purpose where your team won't be negative from start or it can also be done to demoralize the enemy team. Often it can also be used to achieve the opposite effect, for example using an underplayed champion to make the enemy underestimate you. This generally only works in low to mid Elo since most champions are known and considered viable when played by the right person in higher Elo. The champions you'll find on a FOTM tier list are usually either fun to play or played by pros.
The Statistical Tier list usually focuses on the potential or actual strength of a champion while also taking into account how hard a champion might be for a new player. Usually these tier lists are also kind of limited to a certain Elo. Tier lists for different League Tiers will have different champions. You can see for example that Shaco is currently sitting at a rather low win rate in bronze while in platinum and diamond he has one of the highest win rates. This somehow shows that the champion is either difficult to play (effectively) or that it fits better into the high Elo playstyle.
Then there are of course also the tier lists for the different game modes like dominion, twisted treeline and even 5v5 Teamranked on summoners rift. But these are usually harder to find and even less important than the soloqueue or FOTM ones.
The arguably most popular Tier list is provided by Nerfplz (FOTM and Regular) but there are many more.
How should I look at the tier list
Maybe you've already guessed it with the information above but once again: Don't look at them as an absolute viability chart!! Tier lists are a good thing to exist and they also provide some useful information but in the end it's just an estimation of champion strength and not a definitive strength order. There are a lot of players out there (like this one) who managed to get Diamond or even higher with champions that are at the absolute bottom of the tier list.
You should always pick the champions that you can play best not the one that is considered the most viable by a random list on the internet. However the list might help you if you're indecisive about what to play or main.
The tier lists are by far not perfect but not all of them are complete garbage