Explanations

MY TRUE LOVE= (a romantic expression still in use) My girlfriend/boyfriend

PARTRIDGE= A low flying bird (see picture)
Partridges are never seen to be on a tree (not even pear trees), but oh well, maybe that's why this partridge is so special.

TURTLE DOVES= A breed of doves which are dark grey and often used two of them a symbol of love. (see picture)

FRENCH HENS= Any of 4 different breeds of hens originated in France. I suppose they must be "faverolles" here, since they were very popular because they layed many more eggs in winter (or so I read), when they were most needed (see picture).

CALLING BIRDS= Black birds (see picture)
This is an alteration of the word "collie", an old word which means "black as coal".

GOLD/GOLDEN= A gold ring is made of gold, a golden ring looks like gold, especially in colour, but it is not necessary made of gold. Maybe that's why the woman says "5 gold rings" and the man corrects her "5 golden rings", ha ha. Personally, I always learnt "5 golden rings" in this carol, maybe Americans are more splendid in their presents. But actually, the "golden rings" refer to the Ring-necked Pheasant, so they're birds too (sorry).

GEESE= the plural form of "goose" (see picture)

A-LAYING= Laying (an egg). If an animal legs an egg (well, not all animals can), they produce an egg.
In Old English the gerund form of the verb was produced with the prefix A- and the suffix –ING. In modern English the prefix has been lost, but in some rare occasions is still used in poetry or songs to provide for a needed extra syllable, and of course, it survives in old language like this ancient Christmas carol.

A-MILKING= Milking. To milk an animal (a milkable animal, that is) is to squeeze their teats (he he) to get milk out of them. All mammals produce milk, but not all mammals are milkable, so be careful :)

A-LEAPING= Leaping, jumpling.

PIPERS PIPING= A piper is a man who plays "the pipes", an old Celtic instrument still used in some parts of Europe, most famous in Scotland (see picture). If they are piping, they are playing the pipes.

Notice something English people love (especially in the past): alliteration. This consists of repeating the same sound two or more times in the same line. For example:

/S/: seven swams a-swiming
/L/: eight lords aleaping
/T/: two turtle doves
etc.

THE VIDEO PARODY

2 turtle doves= A turtle dove is a kind of dove, but on the video they represent 2 turtles with dove wings, a mixture of a sea turtle and a dove.

3 French hens= They are a just a variety of hens, but here they are hens from France, so they are wearing the typical black and white striped t-shirt associated with the French.

4 calling birds= A calling bird (or collie bird) is a dialectal version of a black bird (a bird which is small and black), but here they represent 4 birds queuing at a phone booth to make a telephone call.

6 geese a-laying= These geese are laying an egg (each), that's why they are described as "geese a-laying" (a- before an -ing form is old-fashioned English). But on the video the represent 6 geese in bed, because if you lay yourself in bed, you are in bed.