Continuity describes whether or not there are any breaks, holes, or jumps in a function. A continuous function has no interruptions and can be drawn without lifting your pen from the paper.
The IVT states that if you have two points on either side of some value within an interval where a continuous function exists, then there must be at least one point within that interval where the function takes on every value between those two points.
A removable discontinuity is when there's initially an interruption in the graph of a function but can be fixed by redefining its value at that particular point.