To evolve is to change through time.
However, the modern definition of 'evolve' is based primarily on biological evolution, grounded in the ideas of Lamarckian Evolution or Darwinian Evolution. The general public's concept of what exactly either type of evolution involves is often rather hazy.
Darwinian, neo-Darwinian, and Lamarckian evolution all are based around the idea that a population will slowly adapt itself to its 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 all be said to evolve.
In general, the use of the word 'evolve' indicates 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 genotypic adaptation over time of a population which better enables it to deal with its environment." The exact definition will depend on what context you are working in; to read more about evolution, you should go to the nodes on Evolution, Darwinism, and Natural Selection. Selfish Genes and The Red Queen Principle are also important ideas.