What does it mean to have a Y-DNA Haplogroup "R-M512"?
Normally Humans inherit from each parent one sex determine Chromosome. If you have a XX chromosome pair you are female. If you have a XY pair you are male. Therefore the Y-Chromosome get only passed on from father to son. The Chromosomes itself consist of DNA. The DNA consists of base pairs in which an A only pairs with a T and a C only pairs with a G. The Y-Chromosome has about 57 million of these base pairs. DNA is normally relative stable but from time to time only one of these base pairs randomly changes from a C-G base pair to an A-T base pair. This phenomenon is called a pointmutation. So imagen two brothers. One has the unchanged Y-Chromosome of their father the other has one this pointmutations. So the descendants of the one Brother have the same Y-Chromosome, with exception of the pointmutation, the same Y-Chromosome. This is called a Single Nucleotide Polymorphisms (SNP). In the millennia the Y-Chromosomes acquire more and more of these SNPs. One combination of these SNPs is called a Haplogroup. By the amount of SNPs and the kind of them you can reconstruct which Haplogroup descended from which one. You have the Haplogroup R-M512 which is a rather old one (https://upload.wikimedia.org/wikipedia/commons/d/d4/R1a._Info-19.05.2014.jpg) and could be found in great area. M512 is actually the name of the last SNP that defines your Haplogroup. Here some people had write down which Haplogroup they belong to. By the ethnicity of the name you can assume how far it is spread ( https://www.familytreedna.com/public/R1a?iframe=ysnp ).
Another Name would be R1a1a. Maybe these papers help you more to understand the nomenclature ( http://www.ncbi.nlm.nih.gov/pmc/articles/PMC155271/  http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1288547/ ).
Short tandem repeats (STR) are small DNA sequences that repeat multiple times.  On the Y-Chromosome these STR can be referred to as DNA Y-chromosome Segments (DYS).  DYS448 has the sequence AGAGAT  with 20 to 26 repeats. Here ( http://cstl.nist.gov/strbase/ystrpos1.htm ) is visualized where actually  it is located on the Y-Chromosome.  In your case the laboratory was not able  to measure any repeats.