Do the numbers and letters on a US vehicle licence plate mean anything in particular or are they just randomly selected to distinguish each vehicle? (For example, here in Ireland the first 2 digits give the year of registration, the letter or letters give the county of registration and the final numbers are random)

License plates in the U.S. are issued by the states and there are variations in the numbering systems they use for them. However, it seems that most states do have some sort of meaning coded into at least a portion of the plate numbers. This page on Wikipedia has some more detail: https://en.wikipedia.org/wiki/Diplomatic_license_plates_in_the_United_States#Numbering_and_lettering

I also want to mention that it is also possible to order a "vanity plate" which is a self-selected, personalized plate number that can be selected by the person ordering the license plate...as long as the vanity plate request is unique and within the character limit of the physical license plate.

well in massachusetts it is completely random. however you may get a plate that is specified to wat you want called a "vanity plate". these cost an extra fee.

I have found that in NY, sometimes but not always, the letters indicate the DMV office where issued. I have seen "A.." plates from Albany, "D...", from Dutchess County, "K.."and "U...", from KIngston or Ulster County.