This method is as basic as a method can be. The Persistence Method banks on that the weather today will be the weather tomorrow. Most people commonly follow this method every day without even thinking about it. If it was 100ºF today, you would probably plan on another hot day to follow. This method works excellent if you live in a region with minimal weather fluctuations during a given season. For example, if you live in Los Angeles (USA), you can probably expect that one sunny day will predict the next sunny day. This method commonly fails in regions where the weather patterns change throughout a season. Many of us, subconsciously following this method, have been caught without a coat because we assumed a warm evening due to the previous warm evenings.
If a nearby region had a storm, you could measure all sorts of weather factors (pressure, wind, cloud cover, temperature, rainfall) in and around that region. With these measurements you could try and predict if and when the weather event will move into your region. The greater your data collection, the greater the odds of success. You can see how this may work for short term forecasts (72 hours), but how the accuracy is quickly reduced the further out you try to predict. As weather factors change, the forecast can be updated but become less dependable to the user.
Let's take that same nearby region that you measured all the weather factors in. Now imagine you did it every day for twenty years. Climatology takes these historical weather trends and develops a statistical average for every day. So if your birthday had a average temperature of 72ºF, for the last twenty birthdays, Climatology would assume your next birthday will be 72ºF. The problem for this trend becomes apparent when you think about all the "unexpected" weather events of the last few years. For this reason many people that depended on Climatology for their planning in the past have now moved onto other resources.
Once again we take the weather factors you measured in the nearby region. We compare it with to the historical weather trends; but, this time we are looking for any days (not just your birthday) that had identical weather factors. If you find a day in the past that had the same exact weather factors, a prediction is made on the assumption that what followed in the past will likely follow in the future. Unfortunately there are a multitude of factors. The result is various predictions using the same method.
Numerical Weather Predictions take advantage of powerful computers able to use complex algorithms. A program is created with a simulated atmosphere. Factors are then plugged in and you see how the simulated atmosphere responds. And while the computers are powerful, and getting more powerful every year, it is the shear amount of factors that cause the problems. Aside from now many factor you plug in, there is the problem with how detailed they are. For example there are only so many weather stations. It is the voids in between that cause the issues.