Battle Pass is a monthly bonus rewards that you can get by simply doing certain tasks. You unlock this at Adventurer Rank 20. The shortcut for this is F4 on PC. You can view the remaining time left on the bottom left of the screen. There are daily, weekly, and BP period missions that you can complete. Doing these will give you BattlePass Experience Points (BEP), so that your Battle Pass level will increase, and you can claim rewards whenever you level up your BP. The basic and free version is called the Sojourner's Battle Pass, and is unlocked by default at the start of each period. If you only have the free version, you can only claim the ones on the top row. If you want to also get the bottom row rewards, you will have to upgrade your BP by purchasing Gnostic Hymn, or Gnostic Chorus using real money.
Unlike commissions that are varied each day, the BP missions are always the same. You can complete the "Mine 10 items" by getting your expedition rewards from the characters that "mined" for ores.
Log in (120 BEP).
Complete 4 Daily Commissions (150 BEP).
Mine 10 items (150 BEP).
Use a total of 150 Original Resin (225 BEP).
Forge 20 items (360 BEP).
Cook 20 dishes (360 BEP).
Spend a total of 500,000 Mora (450 BEP).
Complete 15 Domain Challenges (450 BEP).
Complete Ley Line deposit challenges 20 times (450 BEP).
Collect 100 Mondstadt local specialties (450 BEP).
Defeat 10 Elite Opponents (450 BEP).
Complete Wolf of the North Challenge (675 BEP).
Complete the Stormterror Domain Challenge (675 BEP).
Use a total of 1600 Original Resin (675 BEP)
This BP Period
Make a total of 50 wishes (1500 BEP).
Use a total of 5,500 Original Resin (2250 BEP).
Clear Spiral Abyss challenge 8-3 (2250 BEP).
Gnostic Hymn and Gnostic Chorus
You can purchase the upgraded BP with real money. The Gnostic Hymn costs about $10 USD, while the Gnostic Chorus is $20. The benefit of upgrading to at least Gnostic Hymn is that you get better rewards per level of the BP, and that you can choose a 4 star weapon of your choice (from a select few) when you reach BP level 30. The Gnostic Chorus gets you what the Hymn version gives you, with the added benefit of +10 BP levels when you purchase it, and you also get a namecard style, as well as 5 fragile resins.
4 star weapon of choice
If you upgrade your Battle Pass with the Gnostic Hymn version, or higher, you can claim 1 of these when you reach BP level 30.
You can use primogems to increase your BP level, and it costs 150 primogems each. This can help people catch up, or ensure that you can get to your desired level, if you get the BP late into period.