A scalar field is a scalar quantity defined everywhere in space. For example, temperature is a scalar quantity; by specifying the temperature for every point in the room, you have specified a scalar field.
Before the advent of General Relativity (GR), the force of gravity could be expressed as the gradient of a scalar quantity, and so it may have been considered a scalar field theory (this scalar field is the gravitational potential). Once GR came into vogue, it was quickly realized that gravity was a tensor theory (for the non-math oriented, a tensor is a generalization of vectors), and the gravitation field was described by a rank-2 tensor.
In particle physics there is only one example of a scalar field: the Higgs particle. Although still theoretical and not yet observed, particle physicists hope that the Higgs field exists as it leads to a "natural" explanation for mass.
So the question is, are scalar fields used in GR? The answer is
yes, and here are a couple of examples.
Brans-Dicke Theory
In the early sixties, Brans and Dicke (Physical Review, 124, p. 925, 1961) considered how black hole solutions would differ if Newton's constant, G, wasn't really a constant, but a scalar quantity which could vary with space. In truth, they considered G to be a function of space and time, but considered the spatial variance only in connection with black holes.Now we know the value of G quite well, and so if there is any spatial variation it would have to be weak since solar and galactic dynamics depend on this parameter. As for time dependence, G has to be currently slowly changing for numerous biological, geophysical, astrophysical and cosmological reasons (see, for instance, H. J. Kreuzer, M. Gies, G. L. Malli and J. Ladik, J. Phys. Am.: Math. Gen., 18, p. 1571, 1985).
Dimensional-Reduction
In multi-dimensional theories of gravity (for example, five dimensions), a scalar field is used to describe the size of the extra dimensions. This scalar field has a mathematical equivalence to Brans-Dicke theory (see my list of publications for more info on that).Cosmological Constant and Inflation
When Einstein was considering the cosmological implication of GR, he introduced an arbitrary constant because he found that there were no static-Universe models without it (at the time, Hubble had yet to discover the recession of galaxies which led to our understanding of the expansion of the Universe). Once observation indicated that the Universe is expanding, Einstein retracted this idea. However, the constant was reconsidered in the '70s to introduce the concept of inflation: a brief period of accelerated expansion near the time of the Big Bang to account for several present cosmological observations (such as isotropy and homogeneity). Since then, inflation has gone through many reformations, and it is generally considered that the presence of a scalar field with certain types of "potentials" induces inflation in the early-time dynamics. If the minimum of the potential is non-zero, the minimum value is the effective cosmological constant.
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