Ascending is to increase the maximum level that your character can reach. Every character will always start with a level 20 cap, so you will need to ascend once you reach their current max. Doing this will increase your character's maximum level by 20 for the first time. However, there is an Adventure Rank restrictions to this to prevent players from over leveling their characters above the world level.
The first ascension requires you to reach AR 15, and will increase max level from 20 to 40. Second ascension requires AR 25, and will increase max level by 10. Every ascension after that requires 5 AR levels (30, 35, 40) until AR 45, where you will then be able to level your character to 90.
The same applies to weapons as well.
How to perform ascension
Level up your character/weapon to their current max first, then the level up button will change to Ascend. You need some ascension materials, and some gold if you want to do this. The materials used for ascending weapons and characters can end up being the same. So it's recommended to chose wisely before ascending, or farm harder for every materials.
Different characters have different Ascension materials. However, the leftmost item is almost always the ascension material with a certain element depending on what element your character is.
You can tap on the item to show where you can get them.
Early on, the items are fairly easy to obtain, and you can even buy them from the souvenir shop in Monsdstadt using Anemo Sigils if you're feeling lazy.
Some materials are also available in the game's "shop" from the main menu. This uses Masterless Stardust that you obtain from doing summons from "wish".
How to unlock passive talents
When you ascend, you can unlock, or increase the potential of your skills. You can view this by going to the character screen, and go to the "Talents" tab. They are mostly passives that will either increase damage, or give some kind utility, like adding an elemental attack to 4th hit from the main character's first ascension passive.