The origin of the name "Mallory" is obscure. Tradition has it that it is cognate with the French malheureux and means 'Unlucky"; but it might also derive from a French (or Welsh - see below) place name.
There are at least two unrelated Mallory lines, each with several subdivisions. The major line belongs to Y-DNA haplotype I1 (which suggests, but does not prove, Norman origin) and includes most of the better-known earlier members of the family.
There are enough differences among the extant I1 results to suggest divergent lines going well back into the Middle Ages - as the standard genealogical documentation also indicates.
The secondary line, represented primarily by Peter Mallory of New Haven http://www.geni.com/people/Peter-Mallory/6000000006728053280 and descendants, belongs to haplotype R1b1a2. There is obviously no possibility of a male-line connection between them and the I1 Mallorys. There are enough differences, including a significantly divergent line represented by John Mallory of Goochland, VA (d. 1845), to indicate that this family group also represents several distinct lines.
Where the Bermuda Mallorys (Horatio Mallory http://www.geni.com/people/Horatio-Mallory/6000000006366309307 and descendants) fit in is not at all clear. If they are related to Peter Mallory of New Haven, they belong to one of the branches of the R1b1a2 group - which means they cannot be offshoots of the main (I1) Mallory lines. On the other hand, if they are such offshoots, it's no use looking for male-line ancestors or collateral relatives of Peter Mallory in that direction.
Another famous Malory who cannot be easily placed is Sir Thomas Malory http://www.geni.com/family-tree/index/6000000019678883005 of Newbold Revel. His direct line daughtered out in the following century, and collateral male-line descendants, if any, remain obscure. He has been assumed to belong to the main (I1) line...but it is so hard to locate his ancestors that perhaps he didn't. (This might mean that the antiquarians who "identified" the author of the Morte d'Arthur as a knight from the border-Welsh region of Maelor http://en.wikipedia.org/wiki/Maelor were on the right track after all, and only wrong in that it was Sir Thomas' distant ancestors who had emigrated from there to the Midlands.)