Copyright grants an exclusive right to an original work's creator to copy, distribute and make derivative works from the original work. The law literally answers the question "Who has the right to copy this work?" However, the law allows other people to infringe on the exclusive right if the infringement is fair. This is called the "Fair Use Doctrine." Section 107 of Title 17 of the United States Code sets out four factors to determine if the use of a copyrighted work is fair. Of particular importance on Tumblr is the third factor, which looks at the amount of the total work used and the importance of the portion used to the work. Since Tumblr posts tend to be short, they are often only pieces of a larger work. Reblogging a few lines from a movie script is probably fair use. Reblogging the entire script, isn't.
If the original poster on Tumblr is the copyright holder in a work, you can reblog it on Tumblr without infringing copyright. Under Tumblr's terms of service, anyone who posts something on Tumblr grants Tumblr the right to copy and distribute the work and for other Tumblr subscribers to do so as well on the site. Thus, if someone posts a picture that she took, you can reblog it on Tumblr.
If the original poster did not create the work posted, then you must determine whether she had the right to post it to know whether you can safely reblog it. If the original poster infringed copyright, you infringe copyright when you reblog something. You should determine whether the work is copyrighted and whether or not the original post is fair use before reblogging something. If you do reblog something on Tumblr without permission from the copyright holder, the copyright holder has the same legal remedies against you as against the original poster.
When you reblog a copyrighted work on Tumblr, you must reblog it with the same copyright notices as contained in the original post. Tumblr follows the Digital Millennium Copyright Act and will remove copyrighted content at the request of the copyright holder. The copyright holder can also elect to sue you for infringing her copyright. If the copyright holder sues Tumblr, under Tumblr's terms of service you must indemnify Tumblr for any monetary judgments, costs and attorneys' fees.