
It all began with tail feathers. In 1878, the Morgan Dollar replaced the Trade Dollar with the initial die showing 8 tail feathers on the reverse eagle. Shortly after production began, the Mint altered the tail feathers, superimposing a 7 TF version over the 8 TF. You can believe or dismiss the stories, but the change may have been made because someone pointed out that eagles have one tail feather longer than the other. Dies were altered again to show 7 tail feathers.
Here is a depiction of all three types, compliments PCGS CoinFacts:

Turns out that eagles have 12 tail feathers, not 7 or 8. (Curiously, the 7/8 TF has 12 feathers if you count the doubling.)
The 1878 7/8 variety has two types, strong and weak, depending on the prominence of the doubling. Here’s an illustration:

After everyone agreed to the 7 tail feathers on the reverse, changes then were suggested for the breast of the eagle. One had the 1878 design and the latter, the design of the 1879 reverse. The 1878 has a flat breast; the 1879, a rounded one.
See this PCGS designation:

Now let’s view all of them in one nifty panel to help identification (right click to expand):

You can read a comprehensive article about the number of strikes per variety on PCGS CoinFacts.
As for value, here are current retail/auction prices for various grades, again from CoinFacts:
Values
| VARIETY | XF40 | AU55 | MS63 | MS65 |
| 1978 8TF | $140 | $210 | $525 | $2,700 |
| 8/7 Strong | $100 | $175 | $525 | $3,500 |
| 8/7 Weak | $100 | $140 | $250 | $950 |
| 1878 Rev. 78 | $85 | $125 | $225 | $800 |
| 1878 Rev. 79 | $72 | $150 | $450 | $2,000 |
Be especially careful when bidding on or buying raw coins of these varieties as sellers often misidentify these types.
Here’s a perfect example, an 1878-S (not 1878 Philadelphia) with no doubling on the tail feathers listed as 1878 7/8 TF Strong:

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