By default, whatever you post on your Tumblr blog is visible to everybody, regardless of whether they follow the blog or not. Tumblr does, however, have some privacy options. Provided the blog in question isn't your primary blog, you can protect it with a password; you can also set individual posts to private on both primary and secondary blogs.
Once you protect a blog with a password, nobody else will be able to follow it, although current followers will remain unaffected. Posts from a password-protected blog, however, won't appear on the dashboard of people following it; while followers will be able to see a small preview of the three most recent posts from it on the list of people they're following, they won't be able to read the entire posts or anything else on the blog without visiting that blog's page and entering the password.
In general, private posts are only visible to you, and only while you're logged in to Tumblr. There is only one exception -- if the private post is on a group blog and you've invited other people to contribute, those other people will also be able to see the post.