The term scalar-tensor gravity indicates a theory of gravity if it is based upon both a tensor field and a scalar field, together. (There can also be theories that incorporate a vector field, a term for one such being scalar-tensor-vector gravity). Newtonian gravity can be (fully) described with a scalar field, the gravitational potential: given knowledge of this field throughout some volume, you can derive the force of gravity at any point within that volume. General relativity (GR) gravity is very close to that described by this model scalar field in situations typical for us, but is ultimately defined by a tensor field: the tensor can be described by a 4×4 matrix, and includes influences in addition to the location of the object. The scalar-tensor gravity theories of interest are generally GR, but the effects of gravity also depend upon a scalar field. A way to look at it is that in place of a gravitational constant is value that is a function of some scalar field: the "constant" to use depends upon location. The term scalar-tensor gravity and other terms for theories of gravity are not necessarily distinct: scalar-tensor gravity can be thought of as a gravity theory that happens to include this particular characteristic.
The terms scalar-tensor gravity and scalar-tensor vector gravity are sometimes used for specific models among those that fit these definitions.