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The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1×n covector with an n×1 vector, yielding a 1×1 matrix (a scalar), while the outer product is the product of an m×1 vector with a 1×n covector, yielding an m×n matrix. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices).
On an inner product space, or more generally a vector space with a nondegenerate form (so an isomorphism ) vectors can be sent to covectors (in coordinates, via transpose), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.
In a quip: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".
More abstractly, the outer product is the bilinear map sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1,1)), while the inner product is the bilinear evaluation map given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.
As a further complication, in geometric algebra the inner product and the exterior (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the "outer (alternatively, wedge) product". The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).
Read more about this topic: Inner Product Space
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