The one on the left is for limited (like Acheron or Luocha right now, or Jingliu tomorrow). The one on the right is for regular banner that's hanging there all the time.
It is recommended to pull first on this banner:
Because it guarantees a 5 star in the first 50 pulls. So pull until you get your first 5 star (as you can see it shows "Remaining warps" in the bottom right corner. Within that 50 you will get a 5 star, if you don't get anything before you
will get it on the 50th pull. I got my Himeko within the first 10 so it says that I still have 40, but it is not guaranteed that I will get another 5 star within that, so I stopped).
Once done with this, you can start pulling on this one:
And here just pull whenever you feel like it. You will get plenty of free tickets for this one by just playing. And the good thing about this banner is that once you pull 300 times, you will be able to choose 1 five star included in the banner.
It shows you the progress too, so you can always tell how far off you are. And also this is another reason why it's better to pull on this one after getting the guaranteed 5* from the Starter one. Even if you don't get anything, it still counts as progress towards whoever you want.
Then Limited banner is this one
In top right it shows that I have 4 red tickets for it, while the other one shows 0, because I used all of the regular ones. The pity from this one doesn't count for the regular one and vice versa.
Also, for the pity system. It is guaranteed that if you don't pull a 5* before that, 90th pull will be a 5*. If you pull on this banner and you get a 5* and this 5* is someone else than Acheron, then the next 5* you pull will be Acheron. So if you ever want to make sure that you will get the banner character it's better to save 180 pulls (90 for the first 5* and then another 90 for the guaranteed banner 5*). It's possible that you will get something before that, but if you want to be certain then this is the way to go about it.
It transfers between banners, so lets say you pull on this one and your 5* is not Acheron. Tomorrow Jingliu comes and you pull for her, get another 5* and it will be Jingliu.