Is it possible to copy one custom object from my salesforce org to another salesforce org? How?
I recommend working your way through some Trailhead modules, specifically in this case:
Probably the most common out of the box tools to start with are
Change Sets to move your object setup (metadata) and then
Data Loader to move the actual records (data).
Change sets only work between sandbox and org and will not allow transfer to another instance. You usually need to use Ant. See this link https://developer.salesforce.com/docs/atlas.en-us.daas.meta/daas/meta_development.htm