Run "sudo /sbin/lspci -v" and look for the entry for "Ethernet Controller".
Mine looks like this:
0e:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168B PCI Express Gigabit Ethernet controller (rev 02)
Subsystem: Toshiba America Info Systems Device ff00
Flags: bus master, fast devsel, latency 0, IRQ 47
I/O ports at 4000 [size=256]
Memory at f2010000 (64-bit, prefetchable) [size=4K]
Memory at f2000000 (64-bit, prefetchable) [size=64K]
[virtual] Expansion ROM at f2020000 [disabled] [size=64K]
Capabilities:  Power Management version 3
Capabilities:  MSI: Enable+ Count=1/2 Maskable- 64bit+
Capabilities:  Express Endpoint, MSI 01
Capabilities: [b0] MSI-X: Enable- Count=2 Masked-
Capabilities: [d0] Vital Product Data
Capabilities:  Advanced Error Reporting
Capabilities:  Virtual Channel
Capabilities:  Device Serial Number 36-38-32-35-33-34-33-33
Kernel driver in use: r8169
Note the last line. It mentions the kernel driver used.
You don't download driver programs in Linux. Sometimes, you do need to find the firmware. This is more common for wireless devices than NIC devices. Usually NIC cards are supported and any firmware is included with the kernel.
After running "lspci -v", also run "lspci -n" and post those results as well. The manufacturer's ID may be helpful in locate which driver handles the device controller.
Also post which Linux distro you have.
Post the result of "uname -r". This is the kernel version.