Inner[f, , , g] is a generalization of Dot in which f plays the role of multiplication and g of addition.
Example: Inner[f,a,b,x,y,g] .
Like Dot, Inner effectively contracts the last index of the first tensor with the first index of the second tensor. Applying Inner to a rank tensor and a rank tensor gives a rank tensor.
Inner[f, , ] uses Plus for g.
Inner[f, , , g, n] contracts index n of the first tensor with the first index of the second tensor.
The heads of and must be the same, but need not necessarily be List.
See The Mathematica Book: Section 2.2.10 and Section 3.7.11.
See also: Outer, Thread, MapThread, ListCorrelate.