A 'beneficial natured Pokemon' is a Pokemon that has a nature which increases the potential power of a Pokemon.
So yes, as said, the nature may further add to a strong stat, or boost a weakness; and on the other hand, hinder a stat that you don't care about.
Generally you would want to aim to catch a beneficial natured Pokemon if you are going to use it competitively.
example: You want to raise a tyranitar and have it in your competitive battle team - usually its role is a physical sweeper. It will be using lots of moves that rely on the 'attack' stat, and none of the 4 moves would be a special move such as surf. So in this case, you wouldn't have to be a genius to work out that you would want a nature that strengthens attack, and weakens sp. attack (which happens to be adamant btw.) So an adamant Tyranitar would be classified as a 'beneficial natured Pokemon.
In short, basically, it is a specific breed of a particular Pokemon that will make it better!