Good explanation is given on Wikipedea I think: http://en.wikipedia.org/wiki/Spray_(mathematics)

Look at "Semisprays in Lagrangian mechanics" and "Geodesic Sprays" parts of the article.

You should study the topic of second order vector fields in Riemannian geometry (that is in the tangente bundle). The book

Lang, S. Fundamentals of Differential Geometry Springer, 1999

has a good account of it.

The wonderful little book by Bröcker and Jänich "Introduction to differential Manifolds" contains a good account of sprays on general manifolds. Sprays are a coordinate free way to write a second order equation as a system of coupled first equations. In particular, the geodesic equation can be formulated as a spray.

A more general notion is that of a semispray, which is a special vector field on the total space o the tangent bundle of a manifold. It integral curves are given as solutions of a system of second order ordinary differential equations. For such a system, once you fix a point $p$ on the manifold $M$ and a vector $v$ in the tangent space at $p$ to $M$, $T_pM$, there is a unique curve that at $t=o$ goes through $p$ with velocity (in the direction of) $v$. Then you can identify the semispray (or the corresponding system of second order ordinary differential equations) with this collection of curves that at each point $p\in M$ there is one and only one going to every direction (here is where the term spray can be used, or "gerbe" in French). When the semispray is homogeneous in velocities then you have a spray, if it is quadratic in velocities you have an affine spray. The system that gives the geodesic equations of a (semi-)Riemannian metric gives you an affine spray. Sprays appear commonly in Finsler geometry, while semisprays in Lagrange geometry.

The term of spray was introduced in

W. Ambrose, R. S. Palais and I. M. Singer, \emph{Sprays}, An. Acad. Brasil., \textbf{32} (1960), 163--178.

When it comes to semisprays, there is a geometric setting one can associate to it (or to more general cases of systems of higher order ordinary differential equations)

See, the following paper to have an idea how it started.

D.D. Kosambi, \emph{Systems of differential equations of the second order}, Quart. J. Math., Oxford Ser, \textbf{6} (1935), 1--12.

For more recent treatments of the subject, please see

Bucataru, Ioan; A setting for higher order differential equation fields, and higher order Lagrange and Finsler spaces, Journal of Geometric Mechanics, vol. 5, no. 3 (2013), 257-279

Bucataru, I., Constantinescu, O.A., Dahl, M.F.: A geometric setting for systems of ordinary differential equations, International Journal of Geometric Methods in Modern Physics, vol 8, no.6 (2011), 1291-1327.

Geodesic spray is a notion for every Riemannian manifold, more generally, for every manifold with a covariant derivative on its tangent bundle (affine connection). It means a type of 2nd order ODE where the orbits depend only on the initial direction, not on the length of the initial vector. A geodesic is like a track engraved into a manifold; you can run along this track fast or slow, but you always keep running the same track.

Best regards

Jost