A file in the file system is basically a link to an inode.
A hard link, then, just creates another file with a link to the same underlying inode.
When you delete a file, it removes one link to the underlying inode. The inode is only deleted (or deletable/over-writable) when all links to the inode have been deleted.
Hard links are only valid within the same File System.
A symbolic link is a link to another name in the file system.