Spacetime is three-dimensional space with time as a fourth dimension. For an event, such as snapping your fingers in your living room, its position in space might be described by distances from the floor and two walls, and its time by the time on the clock, i.e., four numbers.
The concept is discussed in relativity because to some extent, what seems like space to one person could be time to another, which is what happens when they are moving in relation to each other. (The effect is too slight to be noticed, but with extreme speeds, becomes significant.) There are still four dimensions, but, in effect, the time dimensions of the two people are at an angle with each other.
Minkowski space is a particular mathematical model of spacetime which fits very well with the Lorentz transformation used in the equations of special relativity. It treats time as a dimension made up of imaginary numbers, i.e., real numbers multiplied by i, the square root of minus 1. The advantage is that measuring distances using the Pythagorean theorem naturally extend to time: whereas a distance across a plane is √(x²+y²) for coordinates across the plane, spacetime "distances" (spacetime intervals, such as between two events at different times) are √(d²-t²) for d as the distance between the events and t as time expressed as (c × our concept of time). This can be expressed as the normal Pythagorean-theorem form √(d²+t²) if t includes i (i × c × our concept of time).
Both space and spacetime conceivably can be curved, and are now assumed to be so in such a slight manner that detecting it is a challenge.