Echo
When obtaining new pieces of equipment or looting Dungeon Passes, you will sometimes find items with a glow around them. And when you mouse over these glowing items, you will see the word "Echoes" along with some lines of text. Echoes are stats that have been added to the item on top of whatever Stats they already have, with the current maximum number of echo lines any one item can have being 5.
Try and fill out your equipment slots with as many great echoes as you can, and challenge even harder dungeons for even better rewards by using echoes!
Line Generation
While the maximum number of echo lines on both equipment and dungeon passes is 6, it is still possible for items to have 0. Sometimes this is because the item is simply not allowed to have echoes, like in the case of grappling hooks. Most of the time however, it's because the number of echoes on an item is always randomly generated from between 0 to 6. Furthermore, the type of these lines and the quality of these lines are also randomly generated from a predetermined list. The types of echoes you can roll on a hat, are different from those you can roll on a bag. The glow around an item's appearance, as well as the color of the item's text, is directly related to the number of echo lines it has and the quality of these lines.
You can find a list of equipment echoes created by people submitting their findings here, courtesy of community member Flim:
Equipment Echoes w/ Calamity Gem
If you are looking for what kind of echoes can appear on Dungeon Passes, then please look no further than the Passes page itself!
Line Quality
Line quality refers to the potential maximum amount of additional stats any one line can roll. When it comes to determining line quality, you can look at the color of the orb next to the line in question.
The colors go, from lowest roll to highest roll:
Gray 
→ Green 
→ Blue 
→ Gold 
→ Ruby (Red) 
If the line is Gray, then that means the amount of additional stats you're getting from it is a lot lower than it could be.
If the line is Ruby (Red), then that means you rolled near to, if not literally, the maximum possible of additional stats for that line!
But what about when you roll a line that you just don't care about, regardless of its quality? What if you find a nice item, but it doesn't have as many echo lines as you'd like?
You can look at the Souls page for help on that :)
Clarifications
Some echoes can be a little difficult to fully understand at a glance, so here are a few detailed descriptions to help with that!
Drop Rate: Increases the drop rate of souls. Increases item drop rate from monsters and chests and rewards, as well as increases your chances of finding more and better echoes.
Echoes Find: Increases the drop rate of souls. Items that drop for you, as well as the items that you craft, have a higher chance of coming with more and better echoes.
Damage Balance: When the damage you deal to a target is being calculated, the lower end of that range is increased, effectively increasing your DPS by making your weak hits not as weak. Currently the best DPS line on bags.
Armor Pen: Automatically reduces the defense of your target by a flat amount. Needs more testing to see just how good this line is, but it's definitely a DPS increase!