The "expansion path" describes how the optimal combination of inputs increase in response to an increase in output, holding all input prices constant. In the case where capital and labor are the only two inputs, the expansion path describes optimal (or equilibrium) capital-labor ratio.
Deriving the expansion path
The economic expansion path is derived by drawing a line through all the points in which the isocost curve and the isoquant are tangent. Each isocost curve represents a fixed level of costs and the isoquant represents a fixed level of output. Therefore, tracing a line through these set of points represents what combination of labor and capital will be used for a fixed level of costs or output.
The points further away from the origin represent higher levels of costs or output. For example, if we assume that the prices for both labour and capital is $1 then the isocost curve that intersects point A can be represented by the isocost equation:
The isocost line that intersects the point B can be represented by the isocost equation:
Thus costs are increasing the further away from the origin we move (the same logic applies to the isoquant). If any of this is confusing, I suggest reading this post about isocost lines.
In this case we see that the inputs (capital and labor) both increase proportionately with the level of costs or output. However, this does not always have to be true. For example, consider the following diagram with 3 different expansion paths:
For the red expansion path, both labour and capital increase proportionately. This expansion path moves along the 45 degree angle.
For the blue bath we see that expansion favors capital. Looking at the expansion path we can see that when capital is 2 units the amount of labor appears to be 1 unit. When the level of capital increases to 4 units, labor only increases to 2 units. An example of a commodity which might follow such a path is agriculture output. At low levels of output, we often observe farmers using very little capital. However, larger farms employ large amounts of machinery (tractors etc.) and use a lot less labor.
Finally, the other expansion path is the green line which represents an expansion path which favors labor. Each time the amount of labor increases by 2 the amount of capital only increases by one unit.