I first tried a LAN/WLAN but then it seemed I needed a network adapter (right?) so I gave up
Um... It's pretty safe to assume that the laptop has a Wireless connection and the desktop has a wired connection.
"Connecting" them, or more precisely, the first step in getting them to communicate, is to connect them to the same network (for example, a router).
my desktop machine and my Linux SVN server is connected to my router via a Cat5 cable; my laptop, my brothers PS3, and his computer are connected via Wireless-N and G.
All of these devices can "see" the others via IP address configurations. Sharing is possible only between my laptop and desktop, as I have taken the time to make them part of the same "homegroup". before this they were part of the same workgroup.
Basically, you say that you "gave up because it needed a network adapter" but they almost certainly both have one.