There are so many things that this could mean, it needs "context," as Ryan W. said. Context is more information about the atmosphere or situation.
For instance, a patient at a doctor's office who said that they were "backed up" would mean that they were constipated. Someone who is working on a computer and says "backed up" might mean that files had been copied. A plumber working on a pipe that is blocked might say that it is "backed up" to indicate that all of the water going into the pipe goes nowhere and eventually "backs up" to the sink, or bathtub, etc.
There are many other examples. It's a common phrase with many meanings. I hope that helps! You chose a good phrase to ask about; people can give many examples.