Added in the 1.6 update, Tears of Dreams in Honkai Star Rail is a Trace and Light Cone Ascension material. Used as a replacement for insufficient path materials, this allows you to level up characters and weapons without too much hassle. This saves you from having to spend hours grinding Calyx for the various items.
TEARS OF DREAMS HOW TO USE Honkai: Star Rail
If you are at Equilibrium Level 1 or above, you will have received some of these materials in the mail. If you haven't, claim them and then explore other methods of getting them into HSR.
At the time of writing, there are only two ways to get Tears of Dreams in HSR. The first way is to get it from the Nameless Glory or Nameless Honor Battle Pass. The second way is to purchase the Herta Contract: Resource Supply V2.
Nameless Honor has both the free and paid track, and you only get the materials from the paid Glory. Keep leveling up and you'll get them at levels 15, 20, 25, 35, 40, 45, and 50. You'll get a total of 312 of these Trace and Light Cone Ascension materials by completing all of these glory stages.