I use Mongo, and so I really only speak Mongo, but your typical web hosting environment will not allow you to create your own database. You want to access the server at the root (admin) level to configure Mongo. To do this, you need something like a VPS or a dedicated server.
However, to just play around with Mongo, I would recommend downloading the binary for your OS and giving it a run. Their JavaScript shell interface is very easy to use.
Hope this helps! Tim
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