A serf was a peasant tied to a lord's land in feudal times. Surf is about waves, riding them on a board, or browsing the internet. Only one is about history: if you mean waves or the web, you want surf.
The core difference
Serf is a person bound to the land in feudal society, while surf relates to waves and casual browsing.
- serf — a peasant tied to a lord's land: Serfs in medieval Europe could not leave the land without permission.
- surf — waves, riding them, or browsing online: She learned to surf in Portugal.
How to tell them apart
Think of the serf as serving a lord: serf and serve both begin with se. Surf, with its u, is what you find on the sea shore, along with the sun and sand.
The two are easy to confuse because they sound alike, and a slip like surfing the web written as serfing the web is a typical error. Serf is a noun only, so there is no serfing in standard writing.