Also known as: 賄賂, わいろ, 袖の下
The exchange of money (or goods/services) for favors (or to simply get stuff done on time/at all), from those without power/authority to those with power/authority. Most forms of it are usually illegal in most places. Usually not a moral/ethical thing to do.
For legal types of such currying of influences, see: lobbying, pork barrel, etc.