The image you linked to is a test tube brush. It's used in laboratories for cleaning test tubes. If you don't have access to a laboratory supply store, you might also try restaurant supply or homebrew stores for bottle brushes, which can fit into a narrow neck but have larger bristles so that you can clean beer or wine bottles easily.
The term 'pipe brush' I've also seen used to refer to a curved brush that's for cleaning the outside of pipes (curved so you can get behind the pipe easily, which isn't particularly useful in this situation.
'Pipe cleaner' tends to be really short bristles, used by tobacco pipe smokers, but you can also find similar things at craft stores (but I wouldn't suggest using the glittery ones for cleaning stuff)
All that being said -- I agree with the Evil Greebo -- you likely want an auger (aka. 'plumber's snake') to clear the clog. Once that'd done if you still have problems, there are some enzyme based drain openers (intended for slow drains) that will help to digest what's left in the pipe to loosen it up. You might see suggestions online of pouring hot water into the pipes, which will loosen greasy stuff, but for kitchen sinks, you'll end up moving the grease further down the pipes as it cools, which can create neighborhood-wide problems, not just a localized problem that can be easily augered out.