Many modern Bible translations are copyrighted and thus place restrictions on the extent to which you can quote or reprint passages from them. Publishers retain rights to their translations because producing a Bible translation is an enormously extensive (and expensive) project. A single translation can represent years of work by hundreds of scholars, theologians, and editors, all of whom need to be reimbursed for their work. By retaining their copyright to the translation and asking you to respect that copyright, publishers are able to pay their translators and ensure continued translation work.
This means that downloading or distributing the entire text of a copyrighted Bible translation is not permitted. Fortunately, most publishers do allow you to quote passages from their translations within a reasonable limit. You can find general copyright information at the bottom of all the passage pages and more complete copyright and re-use information by clicking on a specific Bible version on this page: https://www.biblegateway.com/versions/. For specific information on individual versions, please contact the publisher directly.
If you are looking for a Bible without any copyright restrictions at all, you do have several options. Some older Bible translations are in the public domain and may be freely reprinted, quoted, and copied without any restrictions at all; the King James Version is the most well-known of these. If you have the skills and dedication, there is nothing stopping you from creating your own Bible translation--there are online projects dedicated to doing just that!
To sum up, Bible versions are copyrighted to make sure that translators are fairly reimbursed for their hard work. Most copyrighted Bible versions have relatively lenient quoting rules that should cover most common situations. However, if you're prevented from using a Bible due to copyright rules, you're free to use a non-copyrighted Bible version.