"to fasten (a ship) in a particular location by or as by cables, anchors, etc.," late 15c., probably related to Old English mærels "mooring rope," via unrecorded *mærian "to moor," or possibly borrowed from Middle Low German moren or Middle Dutch maren "to moor," from West Germanic *mairojan. Related: Moored, mooring. French amarrer is from Dutch.
"tract of open, untilled, more or less elevated ground, often overrun with heath," c. 1200, from Old English mor "morass, swamp," from Proto-Germanic *mora- (source also of Old Saxon, Middle Dutch, Dutch meer "swamp," Old High German muor "swamp," also "sea," German Moor "moor," Old Norse mörr "moorland," marr "sea"), perhaps related to mere (n.1), or from root *mer- "to die," hence "dead land."
The basic sense in place names is 'marsh', a kind of low-lying wetland possibly regarded as less fertile than mersc 'marsh.' The development of the senses 'dry heathland, barren upland' is not fully accounted for but may be due to the idea of infertility. [Cambridge Dictionary of English Place-Names]
Hence moor-fowl "grouse" (c. 1500); moor-hen (mid-14c.); moor-cock (c. 1200 as a surname).