Semi-permanent hair dyes were originally invented to cover grey hair. Semi-permanent dyes partially penetrate the hair shaft, and are typically formulated to wash out after 5 to ten shampoos. The reason why semi-permanent dye is good for covering greys is because grey hair has no pigment in it to begin with - on hair that is naturally colored the dye will give the hair a tint or cast of whatever color the dye is, but will not remove the natural color. Most funky/unnatural colors are semi-permanent.
When dying your hair a crazy color, it is important to remember that semi-permanent colors do not lighten hair, they only color it. So, if you start out with dark colored hair, you will need to bleach it if you want a very bright color. See the node How to dye your hair an unnatural color for more info on this.
There are many different brands of semi-permanent hair dyes that come in unnatural colors. I've dyed my hair using a lot of them, and have talked to a lot of people about them. Here is what I have learned:
Manic Panic: Manic Panic was one of the first hair dyes to come in unnatural colors. It is widely available and animal friendly. Most people I know, myself included, do not really like manic panic hair dye. It is runny and hard to apply, and fades really fast, and gets on everything.
Special Effects: Special Effects is a bit more expensive than Manic Panic, but works much better and my personal favorite. Special Effects comes in a much larger variety of colors. It is thicker than Manic Panic, so there is less of a chance for you to drip it everywhere. It also stays in for weeks, and most colors (especially the red based colors) fade really nicely. Special Effects is sold at Hot Topic (but in limited color selection, mostly reds). The best place to get it is to order it online. www.amphigory.com has some great photos of most of the colors in action so you will know what you are getting.
Punky: Punky blues and purples come highly recommended to me by many friends. Punky is about mid-range between Manic Panic and Special Effects in price, color range, and effectiveness. It is available at most Sally Beauty stores.