The 's implies ownership. Here are a few examples:
A) It could be used in the sentence "that is the chemist's car." In that sentence, "chemist" is still a person, "car" is an object, and "'s" implies ownership.
"chemist's car" -> "car owned by the chemist"
B) In the case of "I'm going to the chemist's" there is no object explicit, so we assume it's the chemist's place of business.
"chemist's" -> "_____ owned by the chemist" -> "shop owned by the chemist" because there is nothing else I care about the chemist owning.
C) "I'm going to John's" is another example. John owns many things, but it's most likely I'm going to John's house, or maybe his shop.
"John's" -> "_place to go_, owned by John." -> probably "house owned by John."
D) "Is that Sally's?" he asked, while pointing to a laptop. In this case, the object is specified by the direction of pointing.
"Sally's" + pointing action -> object owned by Sally
