What is giving me trouble is the phrase
+ your Constitution modifier per [class] level after 1st.
Does this simply mean I add my CON modifier once each time I level up, or add my CON modifier a number of times equal to my class level?
Each time you gain a level you will add your constitution modifier and one roll of your hit die to your previous max HP. If a barbarian with 18 constitution levels up he takes his previous max plus a roll of a d12 plus 4 for his constitution modifier.
On a related note if your CON modifier increases later you gain hit points retroactively for all previous levels.
I have to agree that the wording is a bit convoluted or rather short in the class definitions. However, it is clarified in PHB p. 177:
(And where some people may get quite wrongly confused, the example they give is positive and not negative... but unfortunately, it does go both ways.)
The math is pretty straight forward and can be expressed as follow:
There are many reason why your constitution will go up or down and thus your max. HP (and current HP too!) will vary if your constitution varies. It is up to your DM to have you decrease your current HP or not, but it would make more sense for you to lose the same amount in your current HP, although the math can be a bit tricky (i.e. if you current HP represents what you had at level 3, then you should lose your constitution modifier × 3, and not your constitution modified × current level.) For sure, your current hit points cannot be any higher than your maximum.
In my formula, I put a minimum of 1 HP per level which does not appear in the PHB, at least not for HP. But there are various places where that
There is also the Enlarge/Reduce spell in PHB p. 237 which says (very end):
Also, without the minimum, you could end up dead just and only because you lose your constitution (i.e. a really bad constitution of 1 would give you a -5 modifier, so at level 20, you would lose 100 HP from your rolls... and if your maximum was close to 100 [sorcerers and wizards have 1d6 or an average of 4 × level... would 80 HP - 100 HP = -20 HP], you would die right there.)
As a side note, that applies to all effects that make use of a modifier. So you could lose spells you've learn before because your Intelligence decreases. Similarly, and quite important, your Dexterity could decrease and thus does your AC. That is much more common than losing constitution or intelligence (also, you may want to read Feeblemind, PHB p. 239 in link with abilities going to 1, some magical treasures also have such wonderful effects...)
As pointed out by Miniman, some creatures (Vampire, Wright) and spells (Harm) may reduce your maximum number of hit points. In that case, this reduction has nothing to do with your constitution nor your level, however, it can further complicate your HP math.