For stocks, one point equals one dollar. So when you hear that a stock has lost or gained X number of points, this is the same as saying the stock has lost or gained X number of dollars.
Although one point always equals one dollar, the percentage value of one point movement can be different for two companies. Let's consider a simple example: If the fictional company TSJ Sports Conglomerate loses four points, dropping from $12 to $8, it would be experiencing a 33% drop in share price. This is dramatically different from a four-point drop experienced by a company like Cory's Tequila Co., which is trading at $104. If their stock drops $4.00 to $100, this only represents a 3.8% decline.
Do not confuse points with percentages. When you hear someone say the stock dropped 10 points, the significance of that drop depends on how high the share price is.
It is important to note that this refers strictly to stocks, nothing else. People often talk about indexes, bond prices or currencies being up or down X number of basis points, which are different. One basis point is equal to 1/100th of a percent, so if someone says the dollar is up 50 basis points, that means it is up 0.5%.
(For related reading, see: What is the difference between pips, points and ticks?)