What are the Buddhas of the ten directions? What are the applications of this teaching in the Buddhist traditions?
Firstly, what are the 10 directions:
"Specifically, the ten directions are the eight directions of the compass—north, south, east, west, northwest, northeast, southeast, and southwest—plus up and down."
Second, sutras that attempt to list and describe the 10 buddhas. http://www.tientai.net/teachings/dharma/buddha/10directions.htm
The traditional cosmology imagined that there could only be one Buddha in the world at a time. By imagining more universes, there could be more than one Buddha.
I think the particular phrase is one of the rhetorical flourishes you see a lot of in Mahayana-- it's a nice way to say all the Buddha's, everywhere. The various names of the realms and 10 Buddhas mentioned in the Dasabhumika (realms like "Without Distress" or "Skillfulness") and Buddha's name "Virtue of Goodness" and "Practicing the 3 vehicles" are reminders of what the goals of Buddhism are and what is worth emulating.
The notion of Buddhas of the 10 directions is used to depict the first Jewel - Buddha - in some refuge trees. (In the picture below on the left one can see a circle of 10 Buddhas) Some refuge trees instead of 10 might have 3 Buddhas which in turn will depict Buddhas of the three times - Past, Present and Future.
Taken together, one can come across a phrase 'Buddhas of three times and ten directions' which will signify that the Buddhahood potential encompasses the whole spacetime we reside in. Wherever you go in space or in time, there is always a Buddha you can take refuge in.