In backgammon, can you resign, and how can you tell when you should?
Since the edit, there are two parts to the remaining question: "When have you definitely lost?" and "How can you tell when you've probably lost?".  The first is easier to answer than the second, so I'll start with that.

The minimum you can roll on a go is a two and a one (note that a double one would give you four moves).  The maximum is double six.  So (assuming all pieces have passed each other) you can determine if you have definitely lost by assuming you will roll double sixes for the rest of the game, and assuming your opponent will roll two/ones.

It would be really nice to include a formula here that takes pip count, piece count and whose turn it is, and computes if you've definitely lost, but such a formula escapes me.  There are some situations that depend on more than these three variables - for example:

|24  23  22    |24  23  22
+---+---+---   +---+---+---
| O   O        | O       O
| O   O        | O
|              | O


In both these finishing positions the pip count and piece count are the same.  However the maximum number of rolls to finish is two in the left hand picture, but three in the right hand picture.

There are two main metrics of backgammon positions (other than pip count).  These are the Thorp count and the Kleinman ratio.

The Thorp count is calculated for each player as:


Start with the pip count.
Add 2 for each checker still on the board.
Add 1 for each checker on the 1 point.
Subtract 1 for each occupied home board point.
If the player on roll's count is greater than 30, multiply by 1.1. 


There is an article here explaining how the Thorp count can be used to estimate the game winning probability.

The Kleinman ratio is calculated as:

(D * D) / S


where D is the pip count difference of the two players, plus 4 to compensate for the player on roll's additional pips; and S is the pip count sum, minus 4 to compensate for the player on roll.

There is an article here explaining how the Kleinman ratio can be used to estimate the game winning probability.

Finally there is an article here which overlaps the previous two, but which recommends using the Kleinman ratio over the Thorp count assuming you are able to learn a large table of logarithms!