Adaptive Radiation
A radiation refers to the process by which one species rapidly speciate into a number of different species. A radiation can be adaptive or non-adaptive. An adaptive radiation is a type of radiation when new species are formed through selection into new ecological niches. Such adaptive radiation typically occurs after the raise of a key mutation that allows for further specialization such as a mutation causing a change in the Pharyngeal jaw as observed in cichlids (Albertson et al. 1998).
You can learn much more on adaptive radiation in the book The Ecology of Adaptive Radiation by D. Schluter
Species divergence
Species divergence refers to the process by which two existing species diverge through time, either through the accumulation of neutral (or non-neutral but equally selected) mutations or by selection on different trait values.
Species Divergence and Adaptive Radiation
Because of the semantic difficulties behind the concept of species (see here), it is unclear from which point can we talk about species divergence and by which point the observed process of divergence is only divergence within a metapopulation.
Ignoring the eventual detail of the definition of species, and considering species divergence as referring to the divergence between any two lineage (whether or not in reproductive isolation), then it is clear that species divergence is part of the process of adaptive radiation.
To answer your multiple questions directly
1) On some sites, it says that adaptive radiation is a form of species divergence, while on others it says that they are different. Which one is correct?
Adaptive radiation is a little more than a form of species divergence as it includes also the event by which divergences starts.
2) Is the main difference that adaptive radiation occurs relatively rapidly and species divergence takes longer?
mmhhh... not exactly as the concept of radiation also refers to the event allowing the divergence to happen
3) Darwin's finches is used as an example for adaptive radiation, but is it also an example of species divergence?
Yes, indeed! No radiation, without species divergence. On the other hand species divergence will occur between any two existing species regardless of whether they speciated through a radiation
4) Are the finches on the galapagos islands all difference species (one species diverged into many)? Or are they all the same species, just with different characteristics (e.g size of beak)?
The Darwin's finches refers to about 15 (different) species. The different species of Darwin's finches do NOT refer to intra-species variation but to different species. In other words, there is some reproductive isolation between these different species. Again, you might want to make sure you understand the concept of species and you might want to read this post (same as the one linked above) on the subject.
