You do not have to purchase the Nvidia kit to do 3d, but you do need it to use nvidia's 3d driver (and nVidia's 3d profiles). The two alternatives are Tridef (I use this personally, and it works well enough, and supports alot of games), and iz3d (haven't used). I use tridef in an AMD system, which is a more open design than nvidia.
I'm not 100% sure, but I think tridef will require you to use 3d Side by side mode, which renders the two perspectives next to each other at half resolution, and your display will need a SBS mode that takes this image and stretches and sequences it for you. If you were to use an AMD card, you would be able to use frame-sequential output , which doesn't halve your horizontal resolution, but you'll also run into HDMI bandwidth issues since HDMI can only support 48fps@1080p (which gives you 24fps per eye).
As a note tridef is not free, but it does have a 15 or 30 day trial you can use to see if you like it, not sure what the retail price is though, it came with my monitor.
Tridef doesn't require side by side. It actually works with several different types of 3D rendering. However, 3D Vision on a TV is going to be poor. TV's do not support dl-DVI as far as I'm aware, and without DL-DVI, you will be limited to 720p.