Non- AR Cannon half-Orcs ( simply because I don't know specific lore )

To understand a Half-orc, one must first understand the Orc somewhat.  
Orcs are generally hateful, ill tempered, nasty minded , and brutal savage to other races and even thier own.  They usually have a social structure of something akin to Norse barbarians , very war like.   Orcs favor strength above all else.  The Strongest lead. The Strongest survive.  What "strongest" is exactly , is up to interpretation.  
They have tribes, colonies, but no true "civilization" as other races know it.  As such, concepts of art, enlightenment, grace, social engineering , and scholarly aspects are completely lost on the race as a whole.  
 However, Orcs over all are a very fertile lot, and very viral.  It is said the secondary drive of an Orc is to breed, and they will absolutely try to breed with anything at any time.  
 
Because of thier voracious proclivity , many varieties of half orc come about.  

Most think "half orc" means simply half human, and half orc.   Mayhaps because these are the most accepted and common half orcs that can be found scattered through civilizations.  Usually offspring of this nature was spurred by a ravaging Orc, on a human female ... possibly an unwilling union on the female's part.  

Half orc, can also include the Ogrillon , which is half orc, and half Oger.  In this case, the offspring is the result of a much stronger Oger, mating with a female orc.  These are truly monstrous beast, seemingly to get the worst traits of both the monstrous humanoid races.  Ogrillons wind up physically bigger and stronger than the strongest of full blood Orcs, and often take over bands , tribes, and Orc held "lands".  

Half Orcs, very rarely can be half orc, and Half Elven.
Elves over-all tend to bury any poetry depicting when a male elf, fell in love with a orc female.  It seems to insult them, even though ... it was possibly true.  Most half orc offspring of this nature, stem from voracious orc male, attacking an unwilling elven female. These are rarer to see, because most elven women would take thier own lives, before bearing a child derived from an Orc.
Though in the case of Drow, female drow of low stature could be forced to mate with orcs as punishment, or amusement of nobles.  Bearing some to "survive", usually as lowly slaves, labor force ... or eventual sacrifices.

Half Orc-Tiefling varieties are common, where planar rifts and gates are present.  

Due to size differences , of course dwarves , gnomes, and halflings would not survive the long term carrying of offspring of a Orc.  The other way around, is possible , but unlikely as dwarves and Orcs harbor a innate hatred of each other's kind, and the other two are simply too small and physically weak to make suitable mates.

Halforc-Goblin varieties have been seen, but very rarely.  Again the size difference is most the factor, but not impossible if the female goblin is larger in size than others.  Varieties from a goblin male, mating with a voracious orc female is probably far more common , but the "pups" would be weak , and not survive long in Orc culture.

Dragonblooded Half Orcs , this is primarily still more than likely a human-orc offshoot , as in many cases, dragonblooded beings ( half dragons , ect ) , do not manifest mature signs of being so, until later in life.

In any case, what transfers from the Orc lineage is generally an increase in strength, longer living than the common Orc, but still "matures" rapidly and at a young age compared to most.
Natural aggression, and a general disdain for social nuisances, and of course disdain for restricting laws of civilizations are common traits.
Intelligence wise , many simply do not get the proper upbringing to be educated, but this doesn't mean they are always dumb, dull witted brutes. Such is a stereotype , while can be commonly found ... such can be commonly found in just about every civilized race as well.
 Appearance wise , half orcs tend to be taller than most Humans, and wider more muscular and larger in physical size.  Their skin colors can range from that of a normal human or elf , to the darker ebonies of the races , green, greenish grey, grey, and mottled with varying hues of yellow and browns mixed in.  Hair is usually a black color , that greys fast as the half orc ages, but exceptions have occurred, depending on how strong the parents traits are.
Most have tusk, others do not.  All seem to have a sloping like forehead of some sort, and small pointed ears that tend to jut out from the head just a bit, as opposed to say elf ears that lay flatter against the skull area.
Human-HalfOrc, some could pass as full human appearance wise.  Others , not so much.  

Culturally ... Half Orcs have no base "culture" of thier own , and rarely have any civilization comprised of just Half Orcs.  They tend to either be raised in the culture of one parent, or the other.  Each is as unique culturally, as such wide spread variety can allow.

// This was written by me long ago, for PnP games, to explain half-Orcs once they became a player-race and outside of the MM.