Understanding the Other Tests
Autosomal Test (atDNA)
An autosomal DNA test provides information from the great majority of your DNA (the autosomes are the chromosomes other than the X, Y and mtDNA, and contain most of your DNA sequences, and genes).
Although full genome sequencing is not far away, it remains unaffordable for most and autosomal DNA tests usually examine up to around 1 million genetic markers (SNPs) spread across the genome (1 million may sound a lot but there are over 3 billion DNA letters in the human genome, so it's still a small fraction but the most informative sites are chosen).
The markers give information about all your ancestors in recent generations, but once you go beyond about 10 generations back into the past (roughly 300 years) only a small fraction of your ancestors have contributed directly to your DNA: So even if William Shakespeare were your ancestor (born ~450 years ago), you almost certainly inherited no DNA from him.
This can be a bit confusing: you did inherit almost all your DNA from ancestors alive at that time, but there are very many of them (perhaps 10 thousand or more), and you only actually inherited your DNA from a few hundred of them - a small fraction. The others are "pedigree ancestors" but not "DNA ancestors": you could have inherited DNA from them, but did not because of the randomness in the 50% transmission of DNA from parent to child.
Mitochondrial Test (mtDNA)
A mitochondrial DNA test provides information about your female line ancestry only. Mitochondrial DNA is passed on by a mother to her male and female children but only females can pass their mtDNA on to the next generation (males are dead ends for mtDNA).
This test, like the Y-DNA test, provides information about one specific lineage – your mother, your mother’s mother, your mother’s mother’s mother, and so on back in time. Again the amount of information provided varies among tests, but the mtDNA sequence is short (just 16,569 DNA "letters") and so sequencing the whole mtDNA genome is already not very expensive.
An mtDNA test can be used for genealogical purposes to test a hypothesis about recent female line ancestry (perhaps arising from genealogical research) or to look for matches in a genetic genealogy database.
The mtDNA test also provides a haplogroup assignment which may, like the Y-DNA haplogroup, be accompanied by a story and perhaps a "migration" map. We know a lot about the present-day distribution of the mtDNA haplogroups, but it is again much more difficult to make inferences about the more distant past.
Big Y Test
The Big Y test is an upgrade for a male who has already taken the regular 37, 67 or 111 STR (short tandem repeat) marker test. The STR markers are used in a genealogically relevant timeframe to match other men to search for a common recent ancestor and are the type of markers used for 37, 67 and 111 marker tests.
SNPs (single nucleotide polymorphisms) are used to determine haplogroups, which reflect deep ancestry and reach significantly further back in time. Haplogroups are predicted for each participant based on the STR test results, and Family Tree DNA’s prediction routines are very accurate, but the haplgroup can only be confirmed by SNP testing. These two tests are testing different types of DNA mutations.
Different SNPs are tested to confirm different haplogroups, so you must have your STR results back with the prediction before you can order SNP tests. The Big Y test scans virtually the entire Y chromosome to cover in essence all known SNPs. Better yet, the Big Y looks for previously unknown or unnamed SNPs. In addition to defining and confirming the haplogroup, the Big Y test can be immensely informative in terms of ancestral roots.