lodging; boarding place
Refers to the specific place where one stays as a lodger, often a room in a private home or a boarding house.
大学の近くに下宿先を見つけた。
I found a place to board near the university.
下宿先の大家さんはとても親切だ。
The landlord at my lodging is very kind.
Compound of 下宿 (boarding, lodging) and 先 (destination, place). The exact historical derivation is uncertain, but the word transparently means 'boarding place'.