Where can I find on the internet a free ethernet controller driver for
my e-machine model T 2984.  My e-machine did not come with a restore
disk.

According to the eMachines website, these are your system specs:
http://www.emachines.com/support/support_info.html?prodName=T2984

Microsoft® Windows® XP Home Edition 
Intel® Celeron® D 340 Processor
Operates at 2.93 GHz
256KB L2 cache & 533 MHz FSB 
Intel 845GV Chipset 
7.25"w x 14.125"h x 16"d 
80 GB HDD 
AOL 3 month membership included, click here for details 
8-in-1 Digital Media Manager (USB 2.0, Secure Digital (SD), Smart
Media, Compact Flash, Memory Stick, Memory Stick PRO, Micro Drive,
Multimedia Card)
512 MB DDR (PC 2700)
Expandable to 2 GB 
56K* ITU v.92 ready Fax/Modem 
Intel PRO 10/100 Mbps built-in Ethernet 
48x Max. CD-RW Drive; 16x Max. DVD Drive 
Standard Multimedia Keyboard, 2-Button Wheel Mouse, Amplified Stereo Speakers 
5 USB 2.0 ports (1 in Media Manager; 4 in back), 1 Serial, 1 Parallel,
2 PS/2, Audio-In & Out
AC '97 Audio 
Intel® Extreme Graphics 3D
64MB Shared Memory

Windows XP should autodetect and install an included driver. It's a generic card. 

If it does not, you can download the driver from Intel:
http://downloadfinder.intel.com/scripts-df/Detail_Desc.asp?agr=Y&Inst=Yes&ProductID=996&DwnldID=4240

Otherwise you can try one of these Intel XP drivers:
http://www.network-drivers.com/companies/496.htm?acd=3&rvd=5&thx=9&bng=7&o=5
The username/password for the site is drivers/all.

If you're using Windows 98, the driver is here:
http://members.driverguide.com/driver/detail.php?driverid=180038
The username/password for the site is drivers/all.

Hope that answers your question.

try http://www.driverguide.com 
username : drivers
password : all