Moral and approval are the same. The approval bonus tile is used with moral boost buildings. Your empire wide approval rating is baised on your average of your colonies I think and is really just a measure for you to use to guess if you'll be reelected in the advanced goverment types.

I don't know what you mean by popularity but I suspect its also the same as empire wide approval.

i believe popularity is based on your status with all the other races combined. you know, friendly, wary, hostile....

so if your ally with all it should be maxed. of course the game is over then too...

this should be easy to confirm, just ally with one or 2 ai then check the graph the next turn to see if your rating improved.

Approval is how content your people are on a planet-by-planet basis; morale is the overall rating you recieve. Hence, some Planetary Improvements, Super Projects, Galactic Wonders, and Trade Goods affect Morale (your overall approval rating - ie Xeno Spices) or Approval (on the planet on which it is built specifically - ie Virtual Reality Centers). I believe popularity depends on your relations with other races - if you have overall ratings that are lower than the other civilizations (ie being less than Neutral with every race or being at war with a few races might make you the least liked in the interstellar community - similarly, if I am indeed remembering this correctly, then being on Allied or Team relations with a majority of the major civilizations will probably make you one of the most popular). Because having heavy influence over your neighbors will actually detract from the relationship score ("Alarming Influence"), wielding influence over your allies is probably not desirable in this case.