An eSATA connection is used to connect computers to external hard drives, and cannot be used to connect a computer to a television. Connecting to a television requires a display cable made to carry video signals.
SATA connections connect a computer's motherboard to hard disks and optical drives. An eSATA port is a SATA port accessed on the outside of the machine.
The most common display cables used to connect computers to televisions are HDMI and VGA cables. Other cables capable of connecting the two are DVI and Apple DisplayPort. HDMI cables are also capable of transferring sound to the television, but none of the other display standards can do this.