You will randomly find (most in the mines) minerals and artifacts that are of interest to Gunther at the Stardew Valley Museum & Library. You can then donate one of each of those items to the museum and collect rewards after certain number of donations.
Items he is interested in will have the description of it say so.
Once you find an item, go talk to him (the museum is located in the southeast part of town, across a bridge to the east), coose to donate and then place the item in an empty slot on one of the tables. If you unlock a reward you will be notified by a text popup, just talk to Gunther again to get the reward.
Dont think these count towards this achievement but the museum is also missing books. You have a chance of finding them by using the hoe on a certain land tile occupied by what looks like three worms sticking their heads up and wriggling about. Doesnt hurt to keep an eye out, you can get all kinds of loot from them.