1st edition are Pokemon released in their 1st release. 2nd in their second release, and so forth...
For example, Zygarde Forms. Zygarde was first release by catching Cells and Cores in the wild at caves. Zygardes caught in there are 1st edition.
Then, Zygarde forms are re-released in another same event. This time, you can catch zygarde cells and cores in grass maps. All combined Zygarde in the 2nd event regardless if the material used in combining (e.g. Zygarde Partial 1st edition to create a Zygarde complete in 2nd release) will become 2nd edition.
Lastly, Zygardes are released once again in Mystery Box. All Zygarde from Mystery Box are 3rd edition Zygardes.
There is really no difference except that 3rd edition are all in Cherish Ball. 1st and 2nd can be in any ball.