In Business:
A core competency is something that the firm does very well, that distinguishes it from the competition. One of Philip Morris' core competency is marketing ability. One of Walmart's is the ability to replenish its stock. Other competencies include: capability to have and communicate a strategic vision, the ability to design new and innovative products ect.
Firms usually have at most 5 or 6 core competencies to stay competitive. McDonalds has exactly four: real estate, restaurant operations, marketing and global infrastructure. A firm that thinks it has 20-30, has to take a good, hard objective look at itself. In order to be accepted, the competency has to pass the following tests: it must be valuable, rare, costly-to-imitate and nonsubstitutable.
In order to stay competitive, a firm must focus and exploit their core competencies and create synergies wherever possible.