Open-Meteo offers free weather forecast APIs for open-source developers and non-commercial use. No API key is required. Pulls its data from the weather services of multiple countries. 7 day forecasts updated every three hours with a maximum resolution of 2km.
The documentation for Accuweather forecast API service.
MetaWeather is an automated weather data aggregator that takes the weather predictions from various forecasters and calculates the most likely outcome. The thinking is that the combined knowledge of the forecasters will produce a better prediction. Free API but drop them an email to let them know if you're going to be hitting their service more than once a minute.
An online open and crowdsourced weather service. People set up automatic weather stations (which are fairly cheap) and contribute measurements that are aggregated into forecasts. Has an API so you can pull data out of it as well as contribute it: http://openweathermap.org/appid Free accounts are, of course, limited in several ways. You can also get weather maps of various kinds from the service to visualize the forecast data. Forecast data is in XML, JSON, and HTML formats.
There is also an air pollution API: https://openweathermap.org/api/air-pollution
4030 links, including 257 private