When you test with 23andMe or the FamilyFinder test at FamilyTreeDNA, you will receive a very general haplogroup assignment. A haplogroup is a set of numbers and letters that represents a group of people who inherited a certain set of genetic mutations on the Y-chromosome from a paternal line ancestor. A haplogroup is a branch of the human genetic family tree and can be ancient or recent. As more people test, more defining mutations are discovered, creating new branches. These branches tend to correlate with migrations across the world over time.
23AndMe’s haplogroup assignments tend to be connected to fairly ancient ancestors (1,000-4,000 years ago). FamilyTreeDNA Y-DNA haplogroups assigned through the FamilyFInder (autosomal) test can be anywhere from about 800 to 4,000 years old. The SNPs (mutations) that define the haplogroups are not evenly tested on the chips used for these tests. You will not have a match list at 23andme, but you can see the Y-DNA haplogroups, where applicable, of your autosomal matches if you look at their profiles and if they are sharing their information.
A Y-DNA test at FamilyTreeDNA (37, 67, or 111-STR marker test) will provide an ancient haplogroup (1000-6,000 years old), but the main value of the test is the match list. These matches will be represented in terms of the number of genetic differences between the markers tested. The main value of this kind of test is that you will see if anyone with your tester’s surname matches others with the same or similar surnames, which indicates a likely shared paternal ancestor within genealogical time. If the matches are not surname matches, then the shared ancestor likely lived before the wide adoption of Jewish surnames in Europe in the early 19th century. You might also match descendants of a famous family or known rabbinic line, indicating you likely share a paternal ancestor with them, regardless of surname. See this paper for more information about Jewish rabbinic lineages.
FamilyTreeDNA also offers a more advanced Y-DNA test, called the Big Y 700. This test can give you a haplogroup identified with a shared paternal ancestor who was born within the past 100-200 years, depending on whether others have tested who are closely related to you. In addition, you will be able to determine which of the 600 or so known Jewish paternal lineages you belong to, if your paternal lineage has Jewish ancestry.
Y-DNA Haplogroup map: Chakazul, CC BY-SA 3.0, via Wikimedia Commons
Resources for understanding Y-DNA results
When you test with 23andMe, you will receive a very general mtDNA haplogroup assignment, likely thousands of years old. You will not have a match list, but you can see the mtDNA haplogroups of your autosomal matches if you look at their profiles and if they are sharing their information. This information is not very useful for genealogy except to rule out a common ancestor on the maternal line.
FamilyTreeDNA full mitochrondrial sequence provides both the most recent known haplogroup assigment, as well as a match list. However, mtDNA mutates much more slowly than Y-DNA or autosomal DNA, so mtDNA matches often share a common ancestor from so long ago that your relationship will not be traceable. Comparing your mtDNA match list to your autosomal match list––and comparing both of these match lists to your own family tree––is the best way to determine how you're related to your mtDNA matches. One valuable use of mtDNA is to prove or disprove a maternal-line common ancestor. For example, if you believe your great-grandmother belonged to a particular family but can’t find paper documentation of that fact, you can test against a mitochondrial descendant of a known maternal ancestor.
MtDNA Haplogroup map: Maulucioni, CC BY-SA 3.0, via Wikimedia Commons
Resources for understanding mtDNA results