http://en.wikipedia.org/wiki/Color_depthThat should be a good start. Basically, if you go one-bit, you have two colors, one being your transparency. You could use black, or you could use some other color and it's still technically 1-bit. Think like the alien ships in Space Invaders.
2-bit would be a little harder to convey. You're technically allowed 4 colors, one being transparency. But I think if you go 2-bit, you need to stick to black and greys, like Gameboy graphics.
As far as eliminating the grid by making your tiles not, well, tile, you basically need more tiles. Variation tiles to make it so that it's not immediately obvious that there's repetition.
The easiest way to make alternative tiles that will fit together that I've found was posted on this board a while ago, but I can't find it.
Basically, the idea is you should take your original tile, divide it in half both ways, horizontal and vertical, and then flip the pieces around so they're all facing the opposite way, before putting the pieces back together into one tile. From there, erase everything except a 2-4 pixel border all the way around in your new tile. This new "frame" tile will serve as the basis of all your new tiles. Just fill in the middle part such that it is flush with the border you've created, and you can make limitless numbers of tiles that "fit" together, but have unique variation and thus, you eliminate the obvious tiling.
If anyone knows the tutorial I'm talking about, could you please post a link to it? Not only because I suck at explaining things and I probably didn't make any sense, but also because I want to credit the guy that came up with the technique. It's proven very helpful to me, time and time again.
