To start, let’s answer the obvious question: what is a build around feat? Simply put, it is a feat that's versatile enough, utilitarian enough, or powerful enough to build a character around. Generally, a build around feat does at least both of the following things in some compacity:
Changes something fundamental about another ability or basic game function:
Some build around feats change something fundamental about another part of the game A great example of this is the feat Antagonize. Antagonize is allows you to use the Diplomacy skill to debuff creatures who don’t attack you, and allows you to use the Intimidate skill to force creatures to attack you.
Another example is the undine only feat Hydraulic Maneuver. This feat allows an undine to “use hydraulic push to attempt a bull rush, disarm, dirty trick (blind or dazzle), or trip combat maneuver.” The feat also specifies that it may be used with “your hydraulic push racial spell-like ability, your class-granted use of hydraulic push, or any hydraulic push spells you cast.” Hydraulic Maneuver is the quintessential example of a build around feat. It allows for a common, well scaling spell to be used in a more versatile way.
Infinite or reasonably large number of uses:
A build around feat must have unlimited or nearly unlimited uses to be a true build around feat. One example of this is the Greater Drow Nobility feat. The feat line of Drow Nobility, Improved Drow Nobility, and Greater Drow Noble all improve upon the Drow’s racial spell like abilities. Both Drow Nobility and Improved Drow Nobility and or improve spell-like abilities, but in the end the two feats leave you with at will Detect Magic and 2 uses of each ability, not nearly enough uses to be a build around feat. Greater Drow Nobility, however, makes all of your racial spell like abilities at will abilities, and your detect magic ability becomes constant. Being able to use your spell-like abilities at will allows you to build around their effects more efficiently, and ensure that you will always be able to access your “trick.”
An example of a feat that is situationally a build around feat is Stunning Fist. While hard to build around if you are not a monk, Stunning Fist is a built in build around feat for monks. Normally, it can only be used once per day per 4 levels. so at level 8, you can reasonably use your tricks on 2 opponents. As a monk, you can use Stunning Fist and additional number of times equal to your level. While not unlimited, a mid level monk will still have enough uses of Stunning Fist that she will be able to attempt to stun all of the high threat level enemies she meets.
The dangers of being a one trick pony:
One of the dangers of building a character around a single ability is being a one trick pony, essentially only having one thing your character can do. Whether it’s in the field or in combat, one trick ponies can hurt a small party’s ability to deal with new or unusual challenges. Having other resources at your disposal is imperative to being able to move effectively through a game.
This is the beauty of building around a feat. By building around a feat, a player allows for his class, other feats, and class options to be put towards other resources. For example, say Mike chooses to build an Undine around the feat Hydraulic Maneuver. He could build a sorcerer who focuses on blasting, using Hydraulic Push as a way to add battlefield control options. He could also build an unchained rogue with the major magic talent, allowing him to focus on trapfinding and skill monkeying. Either way, his build around feat allows him to add a significant dimension to his character without becoming a one trick pony.