What does "parts by weight" or "parts by mass" specifically mean?
It seems that the unusual wording comes from the fact that we are used to having all of the atomic masses available to us simply because we have measured all of them. In the early days of chemistry, however, things were not so simple. Thus, the equivalent weight (technically mass since it has units of $\mathrm{kg}$ rather than $\mathrm{N}$) was used as a method for determining a weight relative to some predefined standard. In this case, hydrogen.

Some of the simplest reactions are acid/base reactions, and this explains why the scale is measured relative to equivalents (where one equivalent is one mole) of hydrogen. That is, in your quote from the wikipedia page, one mole of hydrogen, which has a mass of 1.008 grams, can react with 8 grams of oxygen or 35.5 grams of chlorine to form $\ce{H2O}$ and $\ce{HCl}$ respectively. Another way of saying this is that every hydrogen atom reacted with two oxygen atoms and every hydrogen atom reacted with one chlorine atom.$^1$ Thus, we see what is meant by the term "parts by mass". Notice this is not the same as the phrase "parts per mass". "Parts by mass" could be stated more clearly as the "parts of X measured by mass". That is, we are indirectly measuring how many parts of oxygen, chlorine, or whatever other X goes into a reaction by measuring the final mass of a substance which reacts with a known mass of 1.008 grams of hydrogen.

The wikipedia article notes that not everything reacts with hydrogen, so in order to find equivalent weights for some materials, you must measure its equivalent weight to a known mass of oxygen and then use the 8:1 conversion to hydrogen equivalent weights.

I hope this clarifies the idea. Don't get worried though, this does not contain any more information than simple atomic weights.



$^1$This is kind of cheating though because we know the weights of oxygen and chlorine, while chemists in the 1800s may not have known this.