World and rules
Time and weather
Day and Night, Clear, Rain and Thunderstorm — one press each. They do what the matching command does, without the command.
Difficulty and rules
Difficulty applies to the world as a whole. Below it is the table of game rules: the rule, and whether it is on.
Asking the server for the current rules means what it says. The panel does not show what it thinks the rules are; it asks the running server and shows the answer. A rule changed from inside the game, or by somebody else, will be there.
Talking to everyone
Message to everyone puts a line into the chat of every player. Maintenance is the same thing with a purpose: say what is about to happen before you stop the server.
Whitelist right now shows who may join at this moment.
How the server looks in the list
Server appearance is the line players see next to your server in their game. Colours in Minecraft are the § sign and a letter after it, painting the words up to the next one; the buttons put them in for you — bold, *italic*, underline, strikethrough, flickering, and Remove formatting to take it all off again.
How it looks in the game shows the result as the players will see it, so there is no need to restart the server to find out.
Next
The settings behind all of this — the file the server actually reads — are in Server settings.