Once you reach adventurer rank 15, you can start ascending your characters. Ascending characters is essentially increasing their current max level. You need to max out their current level, then get ascension materials + some gold for this.
Different characters require different ascension materials, but they are often mostly a combination of Elemental stones, regional specialty materials, and enemy drop materials
There are 6 elemental bosses. 3 in Mondstadt (Anemo, Electro, Cryo), and the other 3 in Liyue (Pyro, Hydro, Geo). Fight the boss of the element that you need the materials for. Simply tap the "navigate" button to find the locations of the bosses.
You can encounter them out in the open like this, which means that you can do multiplayer if you are having difficulty doing this. You unlock multiplayer at Adventurer Rank 16 though.
These bossfights aren't that difficult, but you should still try to be prepared. Get to know how the element system works and use an element that works against them for an advantage in battle!
Once you beat the boss, a Trounce Blossom will appear in the center of the area. You can open it using 40 resins to get the rewards, or just leave it and get nothing (yes, not even exp for fighting the boss).
What rewards you get are random, so you are not guaranteed to get the ones that you want. The boss will respawn in about 5 minutes after you defeat it, so you can try it again, as long as you have enough resin to pick up the rewards.
The ascension for characters at level 40 and higher requires the next tier of elemental stones. You can still get them as drops from the bosses, but you can also craft them by using the lower tier elemental stones. It takes 3 sliver to make a fragment, 3 fragment to make a chunk, and so on. The crafting table is available in Mondstadt near the fountain.
Each character requires regional specialty materials. Characters from Mondstadt will require items only found in Mondstadt, and characters from Liyue will need materials only found in Liyue. These materials refreshes each day, so be sure to mark them on your map once you find them so you don't forget their location.
You can tap the material when you want to ascend your character to see where they can be found.