I don't want to give anything away, so if my description seems vague, sorry. It's a 2D puzzle-platformer with a time dynamic similar to that of Prince of Persia. In an added twist, time controls vary per world, in world 3, for example, there are enemies and items that are immune to your rewind controls, persistently moving forward even as the world around them is rewinding. This is arguably the least-interesting variation, but I'll leave it at that because a big part of the joy of the game is in discovering the gameplay.
Indeed, like Portal, this game is in many ways a prolonged tutorial, gently introducing the player to gameplay elements without ever stopping to tell you how to play. The game opens with a playable title screen, in which you are standing on a bridge, and the player immediately realizes that this is a 2D platformer. Most worlds open with a level called "The Pit" that is essentially a sandbox for you to discover and sort out the particulars of the unique time dynamic for that level. This dynamic is woven thematically into the story of the game--for example, in World 2 the player discovers the rewind function, and the title of the world is "Time and Forgiveness". Being able to rewind time allows the player to accomplish platforming feats that would be cruelly difficult in a standard platformer.
In this way, Braid seems at all times to be keenly aware of itself--almost mockingly. You solve platform puzzles and are reward with puzzle pieces. Plot and gameplay interact thematically, not through cutscenes--for 95% of the it, what you do in the game has absolutely nothing to do with the story, but the concepts being explored in the gameplay are intimately linked to plot themes. One of the key features is time control, so you start the game in World 2 and have to work towards World 1. It is first and foremost a platformer, and it pays more than one homage to the genre-defining Super Mario Bros. Your main enemies (that you defeat by stomping on them) are little brown hairy dudes that bear more than a passing resemblance to goombas. There are chomping plants that come out of pipes. There's a sidequest to collect stars. Also, you're traveling to castles in search of a princess and these castles have flags in front of them.
Incidentally, the flags at said castles are international maritime flags. This is not hugely pertinent; it's there for those who notice. But the flag meanings are part of the story, if you care to look that deep. In fact, there is an incredible amount of depth to the story and it's totally optional. There's the functional story: find your princess. Then there's the narrative, which has a rather startling twist if you pay attention during the final stage. But then the narrative can be interpretted as a metaphor for something else that is hinted at at various points in the game. Again, it's there to add to the richness, but it never gets in the way.
Braid is confounding and philosophical, it's at once a celebration and repudiation of its predecessors, and it's simultaneously unique and familiar. It's gorgeously artistic, and at the same time it's a friggin' 2D platformer. But check it out anyway, because it is one of the most singularly interesting games you'll ever visit.