Unlike traditional markets, the markets that trade in ether (ETH), the native currency of the ethereum network, never close, so the price of ETH relative to fiat currencies (like the U.S. dollar) may change from second to second, 24 hours a day, 7 days a week.
Ether trading doesn't take holidays off, either.
The price of ETH is determined by the price that people are willing to trade it for. Supply (ETH supply is continually growing) and demand (more people wanting more ETH) determine the price.
You don't have to buy a whole ether at once. You can buy a fraction of an ether, and secure it yourself.