Daily forecasts for weather can be divided into three historical methods:
Predict based on what you can see in the sky (and later, what you can measure locally with instruments). This is only useful for short-term forecasting, sometimes only a few hours, and isn't very reliable. The simplest version is "It's getting cloudy and they're looking really dark - I think it will rain soon."
Predict based on local and distant observations. Given weather observations over an area, the movement of storm systems, of warm and cold fronts, etc., can be tracked. Once one knows how weather systems are moving, the weather at a particular location can be forecast. For this to be useful, the information about nearby weather needs to be collected quickly enough for a sufficiently up-to-date picture of weather in the region to be made. The important enabling technology for this was the telegraph, so this kind of weather forecasting dates to the mid-19th century. For what is apparently the first published weather forecast of this type, from 1861, see http://www.bbc.co.uk/blogs/23degrees/2011/08/150_years_since_the_first_uk_w.html (note that it's very vague: "North - Moderate westerly wind; fine."). This method isn't useful for long-term forecasts (more than a day or 2), since it depends on the current movement of the current weather patterns in the area continuing as they are.
Forecasts based on computer models of the weather. The idea goes back to the early 20th century, but the calculations took too long for the forecast to be useful before electronic computers. The simplest versions of this started being used for practical forecasts in the 1950s, but there were lots of improvements over time.
Improvements in numerical weather forecasting have depended on three things:
2, Improvements in theory.
The first two are closely connected. Improved theories are often more complex theories, and produce larger, harder-to-solve mathematical models. There is little point in making complex mathematical models for a 2-day forecast if it takes 2 days to numerically solve the model. Improved computers allowed the use of improved models. Similarly, improved models allowed forecasting to make use of increased computer power.
The third point above, improved data collection, often gets less attention than improvements in computers and theory. This is where radar contributes, with radar providing current information about precipitation and wind speeds, including data above ground level. Satellites also contribute important data. An unsung hero of improved data collection is automated weather stations, which have allowed more data collection points, each reporting their data more frequently, compared with weather stations that required manual data collection - a human would have to go and read the rain gauge, the thermometer, barometer, and anemometer.
Over the last half century, useful predictions have been pushed forward in time by about 1 extra day per decade (Lynch, 2008). Short-term predictions have improved in accuracy, and longer-term predictions have become useful.
One challenge for accurate forecasting is that the weather is a chaotic system (in the mathematical sense) - small errors in the initial data will result in a large deviation between the prediction and reality after some time. The deviation grows exponentially (again, in the mathematical sense), so while the error in the forecast might be acceptably small for some time (e.g., a few days), it can then rapidly grow to the point where the forecast has become worthless. This is one reason why improved data has been very important - one way of reducing the error in the prediction is to start with better data.
Reference:
Peter Lynch, "The origins of computer weather prediction and climate modeling", Journal of Computational Physics 227(7), 3431-3444 (2008): https://doi.org/10.1016/j.jcp.2007.02.034 (it's about the history, not the physics, and is IMO accessible to the non-physicist).