When we say that "space is bent", we mean that the geometry of space is altered from "flat". Flat geometry is known as Euclidean Geometry and is the geometry you were taught in school. Geometry is: a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. (Per wiki) When we talk about space bending we are really talking about how objects behave and move relative to one another within that space. The mathematical way of setting up a model and figuring this stuff out is to assign some sort of value at every location in space. This is what a "field" is. A magnetic or electric field is technically just assigning different values to every point in space that tells you the direction and magnitude of the force a charged or magnetic object would feel at that location. For gravity, this "field" is known as a metric tensor and is much more complicated than an electric or magnetic field.
There's quite a lot involved in describing gravitation and the "bending of space". The current theory that does this is Einstein's General Theory of Relativity. GR is very advanced and very difficult to work with and requires a considerable effort to understand. If you're interested in learning more, I highly suggest starting off with Special Relativity, learning it, and then moving on to GR. Special relativity is the low-energy, low-mass approximation of GR and is typically what people start off with before trying to wrap their heads around GR.