Six years ago, I purchased my Sony Handycam CCD_TRV308 Hi-8. Now I want to download some old videos. Will this cam corder allow me to transfer videos to my computer?

You likely need a video capture card that will accept analog video (several makers like Pinnacle--not a recommend);
...alternatively, if you have a digital video camera, better ones have an analog video input jack by which you can use the digital camcorder as an interface (with analog to digital conversion), and thence to the computer via the firewire cable.
...This technique can also be used to transfer VHS from your old VHS player from its analog output, via the digital camcorder to the PC.
..This way you can transfer into a NLE (non-linear editor) like Adobe Premier, Final Cut, Sony Vegas, AVID, etc ; and edit your video now that it has been digitized. And the Hi8 as you know will have much better resolution potential than the VHS.