Space-time continuum is a description of the space in terms of four dimensions.
If we think of a number line, then there is only a single dimension. And x = 2 denotes that the description x = 2, is 2 units to the right of the origin.
If we think of a two-dimensional space with horizontal and vertial-axes then (2, 3) denotes a point in the space which says that the point exists 2 units to the right of and the 3 units above the origin.
Let's extend this to the 3-dimensional space that we all live in. For example, we can use x and y-co-ordinates to denote our position on the surface and the z-co-ordinate to denote how high we are above the surface.
Let's extend this to include time as well. E.g. let t denote the time in hours since Jan 1, 2017, 12:00:00AM. Then the x, y and z-co-ordinates of the southern most tip of South America do not change but the fourth co-ordinate increases by 24 every day. E.g. if the co-ordinates are (x, y, z, 0) on Jan 1, 2017, 12:00:00AM, then the co-ordinates are (x, y, z, 12) on Jan 1, 2017, 12:00:00PM etc.
This is the space-time continuum. The word continuum is used to denote that x, y, z and t-coordinates do not have any jumps in them. In other words, if x, y and z can only take integer values then this is no longer a continuum.