What is the difference between Milk Chocolate and Dark Chocolate?
Short answer:  yes, milk chocolate differs from chocolate by the milk its manufacture.

Real chocolate (as opposed to many other confections) is made from chocolate liquer, which despite the name, is not alchoholic, or even liquid at room temperature.

The fruit of the theobroma cacao tree contains seeds, known as beans.  The harvested fruits are allowed to ferment, bringing many flavor and chemical changes to the beans, as well as removing the pulpy fruit.   The beans are then husked for the seeds inside, the nibs.

The nibs are the first true chocolate product, although they are not sweet.

The nibs are roasted, and then ground.  This is chocolate liquer, a solid colloid of cocoa fat and solid particles.  It would be solid at room temperature, but the grinding process melts it.

The cocoa liquer can be pressed to separate out the cocoa butter (as for use in the cosmetics industry), leaving cocoa powder, but that is not the point of your question.

Instead, to manufacture chocolate, the liquer is conched, a mechanical process that makes the suspended particles much smaller, part of what gives chocolate its smooth mouthfeel. 

Various products can be made from chocolate liquer, or cocoa butter, including:


baking chocolate - essentially, just chocolate liquer, hardened and tempereded.  May or may not be fully conched, so may not be as smooth as chocolate intended for eating.
chocolate or dark chocolate - Chocolate liquer, possibly extra cocoa butter, and sugar.  Minor optional ingredients often included are vanilla or other flavorings, and lecithen, an emulsifier.
milk chocolate - Same as dark chocolate, with the addition of condensed milk or milk solids, depending on whether it is made via the Swiss method or the Hershey method
white chocolate - cocoa butter, plus sugar and other flavorings
chocolate chips - Another form of chocolate in a particular shape.  Many manufacturers don't make these from true chocolate, but rather substitute another fat which doesn't melt as easily as cocoa butter, for economy, and so the chips hold their shape in the oven
chocolate bunny - Chocolate molded into the shape of a bunny, then tempered and cooled
German's chocolate - A brand name of quite sweet dark chocolate


Chocolate labels which list "cocoa percentage" are saying what proportion of the chocolate is cocoa liquer or additional cocoa butter or cocoa solids--that is, stuff from nibs, as opposed to sugar or other flavorings.  The cocoa percentage for milk chocolate tends to be much lower than that of dark chocolates, although not every milk chocolate has a lower percentage than every dark chocolate.

See this question for information on tempering chocolate, which gives its snappy mouthfeel. 

Edit:  On dairy products in dark chocolate:

I was very surprised at Lemontwist's comment, so I did some googling and found this article at Go Dairy Free:


  A good quality dark or semi-sweet chocolate will only have sweetener /
  sugar in some form added, and may also include a touch of soy lecithin
  as an emulsifier. These brands are milk-free by ingredients, but keep
  in mind that most brands of chocolate are made on shared equipment.
  That is, an inherently milk-free dark chocolate may be made on the
  same equipment as milk chocolate. See below for my note on
  cross-contamination issues. 
  
  The complications arrive as some brands of dark and semi-sweet
  chocolate do include milk ingredients for a “smoother” end result.
  This is particularly true in mainstream brands like Hershey’s. Some
  ways that you may see milk listed in the ingredients include milk
  solids, milk, milk powder, whey, butter oil or butterfat (see the Ask
  Alisa post on butter oil), or even casein. If milk is in the
  ingredients, it should be listed in a clearly identifiable manner per
  the labeling laws, but still, use caution.


This is still in line with the information I provided, as I did mention "other flavorings" in dark chocolate, and it is not a universal or even common practice as far as I know.

For people with strong allergies, the cross-contamination issue may be more of an issue.

Vegans would have more of an issue, as lecithen is a very common ingredient in chocolate of all types, and may be animal sourced.  Vegans would specifically need to reseearch and obtain chocolates that meet that standard.   Googling will find many such products, but I did not find an easy single reference list.