Wish is Genshin Impact's Gacha system, or the summoning feature. You can use Intertwined Fates, Acquaint Fates, or Primogems to turn into the fates to perform a summon.
How to Perform Wish
You can go to the wish screen with a shortcut by pressing the F3 key if you are playing on PC. Otherwise, go to the main menus by tapping on the top left icon and select "wish".
Fates and Primogems
There are 3 different currencies used for the gacha.
The Acquaint Fate is used for the standard banner, and the beginners' wish banner. You can only summon on the Beginners' wish 20 times, while the standard one is permanent. The main difference between the two banners is that Beginners' wish costs only 8 Acquaint Fate and also does not contain any 4 and 5 star weapons, increasing your odds of getting a character.
The Intertwined Fate is used for the limited banner(s), like the one shown below. There is a time limit shown on the banner, so you know that it's a limited banner and it will change to a different one when the time is over.
Primogem is the one that you get in the game when you play and get achievements. This can be used to exchange for either the Intertwined, or Acquaint Fate. It does have other uses like replenishing your Resin too.
Genesis Crystal is the currency that you get when you do an in-app purchase (IAP). It is mainly used to convert into Primogems so that you can convert them into either the Intertwined Fate, or the Acquaint Fate. It can also be used to buy the supply bundles available in the shop section.
There's also Masterless Stardust and Masterless Starglitter. You can get 15 Masterless Stardust if do a summon and get a 3 star weapon.
Masterless Starglitter is only obtainable once you receive a duplicate of a character through a summon. Getting a 2nd-7th duplicate of a 5 star character will give you 10 Masterless Starglitter each, 25 for 8th duplicate and onwards. If it's 2nd-7th duplicate of a 4 star then it's 2 Masterless Starglitter, and 5 for the 8th and onwards.
Genesis to Primogem
The conversion rate from Genesis Crystal to Primogem is 1:1. This means that if you have 300 Genesis, then you can get 300 Primogem. To convert this, you will need to go to the "shop" menu, then tap the plus icon next to the primogem. The important thing to keep in mind is that you can't go from Prismogems back to Genesis Crystals once you convert, so make sure that you're certain of it before you do this.
Primogems to Fates
It costs 160 Primogems to get either of the Fates. This means that if you want to do 10 summons, you will need 1600 Primogems, or 1600 Genesis Crystals. You also can't use the fates to exchange to Prismogems, or with each other.
The base rate for getting a 5 star character or weapon is 0.6%. While the 4 star character or weapon is 5.1%. Getting a 3 star weapon is 94.3%.
The pity system in Genshin Impact is 90 summons for a 5 star, and 10 summons for a 4 star. This will reset if you happen to get a 5 or a 4 star before you reach that amount.
Whenever there is a rate up, there's a 50% chance that you will get the character with the increased rate, only if you happen to get the rarity and weapon/character for that roll.
Example: The chance of pulling a 5 star weapon/character is 0.6%, if you do a summon and managed to get in that 0.6%, you will have a 50% chance of getting whatever is in the advertised banner, or 50% to be whatever is in the pool.
If you want to do an in-app purchase (IAP), you can do so via the in-game shop by going to the main menu, and selecting shop on the top left.
Blessing of the Welkin Moon
Blessing of the Welkin Moon is essentially a monthly premium currency gift. On initial purchase, you will get 300 genesis crystals (300 prismogems since it's 1:1 conversion), and 90 prismogems every day for 30 days. It has the highest prismogems per dollar value, so it can be a good option to consider if you want to top up. Genesis Crystal can also be used to pruchase the bundles in the gift shop. So make sure you check it first and see if you want it or not since you can't convert back to Genesis Crystals if you already exchanged it to Prismogems.
Make sure that you read the limitations of the Blessing of the Welkin by pressing the question mark icon on the picture if you plan on buying more than 1 of it.
This is the section where you get to exchange your starglitters, or stardusts, that you get whenever you summon, for some items. These items get refreshed every month, so be sure to check it out each month.
Crystal Top Up
This is the section for in-app purchase to get Genesis Crystals. The first time you purchase them, you will receive double the amount. Use the customer service button on the bottom right if there's anything wrong with your purchase.