You have three choices:
1) Use a USB connection to a USB, external, case (limited to 32MB/second with unsure Standby capabilities)
2) eSATA (computer) to SATA: This requires an external case with a power connection. eSATA supplies both power and data, but if you only connect to a SATA connection (eSATA to SATA connector) then the power comes from the AC adapter instead of through the cable.
3) eSATA to eSATA. Since you have eSATA on the other end buying a device that supports this is a good idea.
Is it possible to plug in the disk after the computer has booted?
1) eSATA: Yes, it is hot-swappable if it is eSATA supported on both ends just like a USB thumb drive.
2) SATA: Yes, however you will need to enter the Device Manager and click "Scan for New Hardware" which will look, and find, your SATA drive. (Rebooting will also find the drive if it is connected and powered on)
Please note there are very new hard drives called "Advanced format Drives" from Western Digital and possibly others that require a special formatting for full performance. This is for XP. Vista, Windows 7 and later OSX support this automatically. The instructions for setup are included with the drives. I recommend getting one. If you don't format with the software you will lose about 10% of your usable space and the error correction will be 67% as good.
*An external SATA drive is treated exactly the same as an internal one if it has a SATA connection. eSATA is best as it is hot-swappable.