Meaning of "auto" in trans autophosphorylation?
Receptor tyrosine kinases are proteins which float around in the cellular membrane. Upon binding of their ligand, the dimerize (two units form a dimer). This releases the kinase domains and the proteins start phosphorylating themselves over cross (meaning part a phosphylates part b and vice versa). This looks like in this figure below (from here, it is an interesting article):



Besides ATP this process needs no further co-factors so the "auto" refers to the ability of the RTK to phosphorylate itself upon activation. See more about autophosphorylation here.