A Riemanninan metric is an inner product on each tangent space that varies smoothly. An inner product on a vector space V is a positive definite symetric bilinear form. But these guys are in 1 to 1 correspondance with the positive definite quadratic forms on V via B(.,.) --> Q(.), where Q(v):=B(v,v). (Indeed, a Riemanian metric on a manifold is sometimes defined to be a smoothly varying positive definite quadratic form!)
And a (real) quadratic form on V is a map Q:V-->X such that given a basis for V, Q(v) is a second degree polynomial in the components of v.
So a Finsler metric on a manifold M would be, by my guess, a smoothly varying map TM-->R, which, when restricted to a tangent space, is positive definite and, given a basis for the space, is polynomial in the components of its argument, without restriction on the degree of that polynomial, or possibly even more generally, with only the positive definiteness requirement.
Wiki reveals that it is actually a map F:TM-->R smooth only the complement of the zero section (weird) which is positive definite and also has 2 more hypothesis resembling those of a norm.
Note that given a quadratic form Q, Q^½ is a norm.
Still, describing the Finsler metric as "a riemannian metric w/o the quadratic restriction" seems a little akward to me.
