Diplomurkunde and Diplom (Diplomlehrer) in English?
Pre-Bologna process, a German Diplom (which is no longer being awarded, and [see cbeleites's comment below] is mostly replaced now by a Master's degree) was generally the rough equivalent of a US Master's degree, attaining it is one of the requirements to start Doctorate studies. 

Your applicant sent you two documents:


The Diplomurkunde is roughly the Master's diploma in the US. It is called a "certificate" (Urkunde) to distinguish it from the Diplomzeugnisse which also includes the transcript of grades. It is the one that one may consider framing and putting on the wall. 
The Diplomlehrer tells you what subject the applicant received his "masters" in. In particular, he received it in Lehrer, which translates directly to Lecturer Teacher. Or, as we may say, it tells you that he received something like a Master's in education. 


To clear up any doubts, though, you could just ask your applicant to ask the Akademisches Auslandsamt of the place where he received the degree to send a letter attesting to the equivalency of the degree. (That shouldn't be too hard, since now they should've converted to a Bachlor/Master system anyway.)