To change through time.
The modern definition of evolve is based on the idea of Lamarkian Evolution or Darwinian Evolution, although the general public's concept of what exactly either type of evolution involves is often rather hazy.
Darwinian, neo-Darwinian, and Lamarkian evolution all are based around the idea that a population will slowly adapt itself to it's environment. This idea of adaptation over time is applied to pretty much everything; web design, car models, sports, personal philosophies, political parties, foreign policy, and much more may be said to evolve.
In general, the use of the word 'evolve' implies that x is changing; that the change is for the better, at least in one major aspect; and that the change is not strictly planned out ahead of time. In general, development that follows a strict plan with no trial and error would not be called evolution; this is probably the only major difference betwixt the modern usage and the 'developing' definition as it existed in Webster 1913's day. The other definitions given by Webster1913 have pretty much been dropped, although the development of x might involve expanding or "unfolding" (in the metaphorical sense).
That is the informal, everyday-as-applied-to-anything definition of 'evolve'. In a modern technical usage, 'evolve' would only apply to Darwinian and Neo-Darwinian evolution, and might be defined as "a measurable changing in a population's gene frequencies" or "the genotypical adaptation over time of a population which better enables it to deal with it's environment." The exact definition will depend on what context you are working in; to read more about evolution, go to Evolution, Darwinism, and Natural Selection. Selfish Genes and The Red Queen Principle are also important ideas. Surf the softlinks, there's lots of evolution out there...