I read stories such as http://onemileatatime.boardingarea.com/2016/11/30/flyingblue-promo-awards-december/ . Now I understand that such stories are something to do with flying the same airline or alliance over and over again and you get points which can be redeemed for a ticket to some place. Are points and awards the same thing, or do they mean different things?
Points are currency handed by the airline. Awards are what you buy with that currency.
As a member of an airline frequent flyer program, you earn points for certain actions. Flying on a paid ticket is the most common example but you can also get points for purchasing or getting services from partners, such as renting cars with Avis to get American Airlines AAdvantage points. You can even get points for things that do not cost money such as watching sponsored videos or filling out some surveys. One thing that you do not get points for is flying on an award ticket.
With accumulated points, you can purchase awards, including an award ticket on the airline issuing the points or on one of its partner airlines. For example, Delta is member os SkyTeam, so if you get points from Delta you can also use them on AeroMexico. You can also buy non-flight awards which can be hotel stays or even items for purchase. The amount of points an award costs is published on the website of the frequent flyer program.
Keep in mind that most frequent flyer points expire after a certain time of inactivity. For example, this is 18 months for American AAdvantage points. Delta Skymiles on the other hand do not expire. To reset your last activity date, you must either earn or spend points. Some partner credit cards prevent points from expiring.
"Points" or "miles" are the currency you earn in the frequent flyer program as you travel. They are deposited to your account if you include your frequent flyer number on your reservation when you travel. They can also be earned by other actions, such as using a credit card associated with the loyalty program, staying in a hotel that offers frequent flyer miles, renting a car from a company that offers frequent flyer miles, and other earning opportunities you'll find listed on the program's website.
Awards are the things you "buy" with your miles. The most prominent such item are "award tickets," where you exchange a certain number of miles for a free flight ticket (in many cases, especially with international flights, you still have to pay taxes and fees, which may be substantial). There may also be "upgrade awards," where you redeem miles for an upgrade to first or business class. And many programs have non-travel awards, where you can spend your miles on consumer goods, magazine subscriptions, and other items. All of these will also be listed on the program's website. Some people become rather obsessive about maximizing their "bang for their buck" and getting the most value out of their miles.