Working Definitions

For the purpose of understanding, here are some working definitions:
A natural hazard is a natural process or event that could be a hazard to people or their property. These processes are a part of nature, and happen whether humans are there or not. They are only considered hazards when they are perceived as a threat to humans (ie. when humans get in their way). Some examples: floods, hurricanes, snowstorms, landslides, volcanoes, earthquakes, tornadoes, naturally-occurring wildfires.
A natural disaster is when one of these processes or events occurs in a localized area over a relatively short period of time. Factors involved in natural disasters include: loss of human life, human injury/displacement/relocation, a state of emergency is declared, or sometimes international assistance is required. Some examples include: many earthquake events such as the 2014 Napa Earthquake, many individual tornado events, and many wildfire events.
A catastrophe is a large-scale disaster that often takes years of recovery. Factors in catastrophes include: greater loss of human life, severe human injuries, millions of dollars in economic losses, permanent relocation/refugees from affected area, a state of emergency is declared, and sometimes international assistance is required. Some examples, the 2004 Sumatra Earthquake & Tsunami, the 2011 Tohoku Earthquake & Tsunami, Hurricane Katrina 2005.
Usually the main difference between a disaster and a catastrophe is the scale of the event(s), numbers of casualties/persons affected, orders of magnitude (for storm and earthquake events), and duration of the aftermath. Often economic considerations are also considered, both short-term and long-term.
References: Keller, E. DeVecchio, D (2015) Natural Hazards: Earth's Processes as Hazards, Disasters, and Catastrophes. Pearson, New Jersey, USA. FEMA, Disaster Declarations site, available at: https://www.fema.gov/disasters/. National Centers for Environmental Information, Climate Monitoring, Tornadoes. Available at: http://www.ncdc.noaa.gov/sotc/tornadoes/201607.