Code:
lspci -knn
.....
03:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd.
RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 06)
Subsystem: Foxconn International, Inc. Device [105b:0d81]
Kernel driver in use: r8169
Kernel modules: r8169
this command shows ethernet card specification and it's vendor ID and device ID. Using these IDs (in square brackets) you can find which driver you need.
there was a fine website kernel-seeds.org, where you could find which driver you need for particular vendorID/deviceID, but seems this site is down now...