Well, you can do vectors in Photoshop (PS) too, not just Illustrator (Illy). However, logos should definitely be made in vector format, especially for a client. However I get the feeling that xsaber has only one purpose for this logo, his forum, and so it's probably not going to benefit him a whole lot to do all of that extra work.
Greyscale is simply a monotone graphic, whereby it only consists of varying shades of grey - no colour. The reason for testing in such a manner is that companies can often use a logo in many different circumstances that might not have been considered at first. For example, the coloured logo might be printed on a black and white printer and so if the logo doesn't look good in black and white then it's simply a bad design that isn't fit for purpose. But as I've already mentioned, I don't think any of this really applies to you.
Unfortunately the legibility of your logo is very poor, especially the first and last characters. It looks like nech. More importantly, PS effects, such as drop shadows, are very rarely used in professional logos, because it looks old fashioned and cheap.
If you want to learn graphic design then I suggest that you read up on some tutorials, they have some great resources at tutsplus:
http://psd.tutsplus.com/