Ludicrous things are funny, absurd, or nonsensical. If someone says something silly or far-fetched, you could say "That's ludicrous!"
Ludicrous originally meant something that was funny, playful, or joking: a ludicrous comment was just a really funny comment. Over time, ludicrous took on a more negative flavor. Now a ludicrous statement might be funny, but it's also ridiculous, hard to believe, off the wall, or even stupid. When people say "That idea is ludicrous!" it usually means the idea could never happen: the notion is laughable. Saying Neil Armstrong was the third President would be a ludicrous statement.
