Existence is the state of being alive or being real. For example, you and your best friend disagree about the existence of Bigfoot if you think it's real and your friend doesn't.
The noun existence can be used many different ways, but it always has to do with being alive or with simply "being". You can talk about the survival of something as its existence, like your yo-yo club having lost so many members that its existence is in danger. The particular way you live is another meaning of existence; if you live a quiet, lonely life in the country, you could describe it as a solitary existence.