In cladogenesis, one species begins to evolve and split into new species. A great example is Darwin's finches. Over time, a finch ancestor split into finches that evolved different beaks to consume different types of food (this allowed them to exist together without competing for the same resources).
In anagenesis, a species evolves through mutations and other means to form a new species. In anagenesis, there is no split like there was with the aforementioned finch example. The horses we see today evolved from much smaller horse-like animal. In that case, there was no split--the horses simply evolved over time through genetic modifications. If you want another example of anagenesis, do a search for the peppered moth. It's an interesting story!
