An option is the choice to continue the deal. So a player option means the player can force the deal to be continued another year (assuming the money is guaranteed, they'll get paid another year... usually used by players who aren't going to find any better contracts), and a team option means the team can force the deal to be continued (the player must stay another year, usually used on players who would be worth more in free agency).
There's also early termination options, which for all intents might as well be player options really. Just means the player can leave early (or choose to play the final year).
Restricted free agency is similar to a team option... except that they must match any other bids (or let the player go). But they still can ensure you are kept by agreeing to the value others offer. Coincidentally, in many (all?) cases, the team a player is leaving can offer MORE money than any other team can in free agency. So the system is slightly slanted to the team that drafts a player (well, quite a lot, when rookie salary structure is considered). But not enough to prevent the superteams of recent!
It's humorous you'd hear it a lot this year... because now would be the one time in history more than any other when player options should basically never be picked up (unless the player's ability has fallen a ton)!
There was a 35% increase in money designated to go to players this year... and as you probably saw, that money goes into huge salary increases all around as teams try to spend what they must (see the reactions) (in fact, teams HAVE to add at least about $14 million to their salaries for this year, going from a cap of $70 mill to a minimum payout of $84 mill).
(* Basketball Reference suggests 20 teams haven't met this minimum salary as of right now..... they would end up doling the leftover money out among their current players evenly if they don't get to this $84 mill salary floor by season's end [teams like Miami are almost $30 million short right now!!! Equal to a $2-3 million "bonus" per player... all for players value-wise tending to be underskilled and on bad teams! A bonus for putting up with being on a bad team, that's a first!])
But imagine you're a player... you can stick with what you were getting paid last year... or get around 35% more in free agency, basically for free (because everyone has more money and it's there to be spent). And if things aren't lined up right for you this year with the right teams, you could just sign a one year contract with someone, at a significant raise, then try to hit the bank again next year when it's all supposed to rise another 8%.
This nba.com story shows a paltry 3/29 (10%) of players took their option [and one was Tim Duncan, who retired afterwards... the others: Caron Butler and Mo Williams, both aging guys who played small minutes last year, and might've struggled to get signed at all (Butler was since waived)]
3/6 (50%) of team options were picked up. Because it's a deal when you can still pay last year's price.
25/37 (68%) of restricted free agents had their contracts matched (I guess if you have the great excess of money, and like a player enough to have played him last year, you're probably going to pay to keep him generally).
Long story short, not a year for seeing player options exercised. And probably won't be for the next three offseasons either, as players with long-term contracts that are across-the-board 40%+ under market value end, and they'll want that same raise the rest of the league has gotten!
Here is a list of upcoming options at Spotrac.
Looking at Duncan, Williams, Butler, and Durant (yes he really does have an option next year... and so he'll almost certainly be opting out... and we could be seeing all this again in a year... to go with Curry, James, Griffin, Westbrook, Paul, Lowry, Ibaka, Rose, and Wade), it seems there's no set number or percentage of how much an option goes up, but indeed it can't go down. Most follow the trend of their contract for the first 4 years... go up a few % each year. But up to how teams set up the initial deal.
Also came across http://www.cbafaq.com/salarycap.htm, which has an absolute PILE of salary details that blow the mind!
According to that, can only put one if player option, team option, or early termination in a contract, and only for final season.
And finally, coincidentally, turns out, though you don't hear about them nearly as often (commonly due to other big buzz terms more unique to each game), the NFL (I hear franchise tag morel and MLB (I hear arbitration more) have option contracts as well. And apparently they're showing up more often in baseball too.