first store; original store
Refers to the very first branch or location of a business, often in contrast to later branches.
This store is the very first branch of that famous chain.
The original store is still operating at the same location as when it was founded.
一号店 is a more common and slightly less formal way to say 'first store'. 第一号店 emphasizes the 'very first' nuance more strongly.
Compound of 第一 (first) + 号 (number) + 店 (store). A straightforward modern formation.