I have often heard that there are more than 4 (3 space and 1 time) dimensions of spacetime. What are the theories that say so, and how many does each predict? Has any experimental evidence been conclusive? And how are these extra-dimensional theories to agree with our daily observations?
We are not really sure.
Classical theories are based on 4 total: 3 for the observed spatial directions, and 1 for time. However, string theory introduces a whole collection of other possible numbers. Some commonly used ones include 10, 11, or 26. Those extra dimensions are supposed to be "compactified" to explain our inability to observe them.