Answering the system-agnostic tag I present you Empire Weather
The simple charts provide a glossary for seeding descriptions, accounting for prevailing weather, and an instant weather option as well. All using a single d10.
Belated Update... h/t @ryanfaescotland
Eight years and counting. I saw a recent upvote, and Ryan's comment. Yes I'm still using this chart. As recent as last week in my current D&D 5e in Greyhawk campaign (entering its almost fourth year!)
Dave Graffram simply makes it direct and with a dash of spice to make it setting relevant. IRL I had the opportunity to take a mountain hike that started off partly sunny. In the first 10 minutes of the hike, thunder. The guide said something witty about bear caves and trees being lightning rods. 30 minutes later, a thunderous downpour that lasted for the entire hike. Unprepared, the entire group was soaked to the skin. The trails became treacherous: roots and rock, covered in rain soaked bark and mosses, provided no perch for shoes. The hot day turned cool in the cloud cover and rain, then humid after the downpour.
I made a comment to my players in the following session about these experiences providing a frame for describing weather in the game, and justifying a liberal use of the Exhaustion rule in overload travel.
I tend to use something like Tarot cards for things like this. One to three cards for a location, major npcs or sometimes even player characters. One just to give a general feel of what might happen. Different decks normally have slightly different pictures. Sometimes looking at the card will give me inspiration, sometime the reading of a card. The suit of the Minor_Arcana can have many readings for example if you get Queen of Cups, may be a local Bishop comes to town; followed by 4 of Coins, he buys a local farm and Seven of Swords, but now he regrets his action (I wonder why?).
Best Answer
As with anything, usage varies, but usually when people say "sandbox" today they mean a campaign that does not have a specific prescribed storyline, but one where the GM sets up a world (or at least a small section of one) and the PCs are free to wander where they will and find adventure where they will. It's about freedom of player choice.
Pure sandbox play is purely simulation driven. A super hardcore sandboxer places a dungeon (or whatever) in the game world and that's where it is, for the PCs to come across or not (and for NPCs to come across before them or not). If a thief is sneaking into a mansion, in a sandbox game he is able to avoid guards and traps, and not have predetermined plot points presented to him regardless of his actions.
Sandbox is not an antonym for adventure module. Some of the early modules, most notably Keep on the Borderlands, were extremely sandboxy, as were many of the early dungeons (Castle Greyhawk, etc.). Here's a place, there's fell monsters and treasure there, go do what you want. More recently, Paizo did a sandbox-style adventure path called Kingmaker for Pathfinder. Sandbox is a different approach from story-driven - a "story of what happened" may emerge from a sandbox session but a preconception of story, or what "the GM wants to happen," is never applied to the game. Adventure paths, being a series of adventures, can try to be sandboxy but generally try to provide enough story to get PCs from one chapter to another, but event timelines and things like that can serve that purpose without being railroads (though people often complain and call things like that railroading, just because they feel pressured to do something).
Railroading, the antonym of sandbox, is simply extreme constraint of choice. Some perceived constraint of choice is always there in any simulated world in that there are always choices that are impossible to physically perform or clearly undesirable, but where you cross the line to railroad is when these things are obviously being imposed by the GM/metagame (usually in the name of "The Story" or "The Plot").
You can be apparently providing a sandbox but using the game world to provide so many restrictions that you are effectively railroaded into a single course of action. A dungeon full of one-way doors that inhibits all teleportation and divination, for example.
Most games are somewhere on the continuum between pure sandbox and railroad, or even move between the two based on need and GM inclination. Many campaigns switch back and forth between railroad and sandbox. Railroading to move the story on when the players lose momentum and sandbox otherwise it a frequent GM tactic that lets the players be free when they want to be but gives them structure when they're feeling lost.
Sandbox gaming can be desirable because it produces a sense of game world reality that enables the player to focus less on the metagame and immerse in their character and the game world. It can be problematic because players can feel like they are spinning their wheels and wasting limited leisure time without more guidance, and because sometimes a preplanned story cam have more "big, interesting" things happen in it plot-wise than a sandbox.
I tend towards sandboxy play, but in my most recent campaign I had players get frustrated and ask for more direct guidance from me on "what they should do" - I am normally reluctant to do that but did so to make them happier. Often players want the illusion of sandbox and unlimited choice, but with the GM pulling strings behind the scenes to keep them headed towards interesting things.