why is price elasticity always negative?

Answer 1

Elasticity is defined as the percent change in quantity over the percent change in price (or maybe the other way around).

When the price rises, people usually want less of it, so there is a negative relation between quantity and price.

You might ask yourself why it is that this relationship is negative, and would you in fact buy more of a quantity if it's price rose. Assuming that you are rational, that is.
That's the kind of thing that economists spend millions of hours on.

Answer 2

It's not. In cases involving conspicuous consumption or superior goods, it may be positive. Generally, though, the negative elasticity reflects the fact that other things being equal, higher price discourages at least some consumers from buying.

Answer 3

The previous answers are good, but here's another way to think of it: the more expensive something gets, the more money you can save by looking for a cheaper alternative. You might like strawberry jam better than grape, but if it costs three times as much, you can switch to grape and spend that extra money on more jelly, or something else. If all jelly gets very expensive, you're more likely to try other sweet condiments, like honey.

This is why very broad, necessary product categories (like meat) tend to be less elastic than narrow categories (like Angus beef). People are more willing to switch to another very similar product (chicken or pork) than they are to look for an alternative to all meat (soybean or tofu). If all meat is very expensive, people will still buy it, but if Angus goes through the roof, other kinds of meat will be substituted.